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Trace (linear algebra)
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<h2 id="definition">Definition</h2> <p>The trace of an <i>n</i> × <i>n</i> <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> A is defined as<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>: 34 tr ( A ) = ∑ i = 1 n a i i = a 11 + a 22 + ⋯ + a n n {\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{n}a_{ii}=a_{11}+a_{22}+\dots +a_{nn}} where <i>aii</i> denotes the entry on the i th row and i th column of A. The entries of A can be <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, <a href="/facts/Complex_numbers/2w7mqI7e">complex numbers</a>, or more generally elements of a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> F. The trace is not defined for non-square matrices. </p> <h2 id="example">Example</h2> <p>Let A be a matrix, with A = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) = ( 1 0 3 11 5 2 6 12 − 5 ) {\displaystyle \mathbf {A} ={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}={\begin{pmatrix}1&0&3\\11&5&2\\6&12&-5\end{pmatrix}}} </p><p>Then tr ( A ) = ∑ i = 1 3 a i i = a 11 + a 22 + a 33 = 1 + 5 + ( − 5 ) = 1 {\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{3}a_{ii}=a_{11}+a_{22}+a_{33}=1+5+(-5)=1} </p> <h2 id="properties">Properties</h2> <h3>Basic properties</h3> <p>The trace is a <a href="/facts/Linear_operator/Q1PBKNVV">linear mapping</a>. That is,<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> tr ( A + B ) = tr ( A ) + tr ( B ) tr ( c A ) = c tr ( A ) {\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} )\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} )\end{aligned}}} for all square matrices A and B, and all <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalars</a> c.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>: 34 </p><p>A matrix and its <a href="/facts/Transpose/8wmsagGS">transpose</a> have the same trace:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>: 34 tr ( A ) = tr ( A T ) . {\displaystyle \operatorname {tr} (\mathbf {A} )=\operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\right).} </p><p>This follows immediately from the fact that transposing a square matrix does not affect elements along the main diagonal. </p> <h3>Trace of a product</h3> <p>The trace of a square matrix which is the product of two matrices can be rewritten as the sum of entry-wise products of their elements, i.e. as the sum of all elements of their <a href="/facts/Hadamard_product_(matrices)/NjAUd4o4">Hadamard product</a>. Phrased directly, if A and B are two <i>m</i> × <i>n</i> matrices, then: tr ( A T B ) = tr ( A B T ) = tr ( B T A ) = tr ( B A T ) = ∑ i = 1 m ∑ j = 1 n a i j b i j . {\displaystyle \operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\mathbf {B} \right)=\operatorname {tr} \left(\mathbf {A} \mathbf {B} ^{\mathsf {T}}\right)=\operatorname {tr} \left(\mathbf {B} ^{\mathsf {T}}\mathbf {A} \right)=\operatorname {tr} \left(\mathbf {B} \mathbf {A} ^{\mathsf {T}}\right)=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ij}\;.} </p><p>If one views any real <i>m</i> × <i>n</i> matrix as a vector of length mn (an operation called <a href="/facts/Vectorization_(mathematics)/P9a2895l">vectorization</a>) then the above operation on A and B coincides with the standard <a href="/facts/Dot_product/tNz8MLjT">dot product</a>. According to the above expression, tr(A⊤A) is a sum of squares and hence is nonnegative, equal to zero if and only if A is zero.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>: 7 Furthermore, as noted in the above formula, tr(A⊤B) = tr(B⊤A). These demonstrate the positive-definiteness and symmetry required of an <a href="/facts/Inner_product/HoyjElEy">inner product</a>; it is common to call tr(A⊤B) the <a href="/facts/Frobenius_inner_product/E8DB5N9E">Frobenius inner product</a> of A and B. This is a natural inner product on the <a href="/facts/Vector_space/M1pxMLx2">vector space</a> of all real matrices of fixed dimensions. The <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> derived from this inner product is called the <a href="/facts/Frobenius_norm/UJvsb9fF">Frobenius norm</a>, and it satisfies a submultiplicative property, as can be proven with the <a href="/facts/Cauchy%E2%80%93Schwarz_inequality/poMzQUIY">Cauchy–Schwarz inequality</a>: 0 ≤ [ tr ( A B ) ] 2 ≤ tr ( A T A ) tr ( B T B ) , {\displaystyle 0\leq \left[\operatorname {tr} (\mathbf {A} \mathbf {B} )\right]^{2}\leq \operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\mathbf {A} \right)\operatorname {tr} \left(\mathbf {B} ^{\mathsf {T}}\mathbf {B} \right)\ ,} if A and B are real matrices such that A B is a square matrix. The Frobenius inner product and norm arise frequently in <a href="/facts/Matrix_calculus/10vKbMg5">matrix calculus</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>. </p><p>The Frobenius inner product may be extended to a <a href="/facts/Hermitian_inner_product/HoyjElEy">hermitian inner product</a> on the <a href="/facts/Complex_vector_space/M1pxMLx2">complex vector space</a> of all complex matrices of a fixed size, by replacing B by its <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a>. </p><p>The symmetry of the Frobenius inner product may be phrased more directly as follows: the matrices in the trace of a product can be switched without changing the result. If A and B are <i>m</i> × <i>n</i> and <i>n</i> × <i>m</i> real or complex matrices, respectively, then<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a>: 34 <a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p> <p> tr ( A B ) = tr ( B A ) {\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} )=\operatorname {tr} (\mathbf {B} \mathbf {A} )} </p> <p>This is notable both for the fact that AB does not usually equal BA, and also since the trace of either does not usually equal tr(A)tr(B).<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> The <a href="/facts/Similarity_invariance/cqm16x0W">similarity-invariance</a> of the trace, meaning that tr(A) = tr(P−1AP) for any square matrix A and any invertible matrix P of the same dimensions, is a fundamental consequence. This is proved by tr ( P − 1 ( A P ) ) = tr ( ( A P ) P − 1 ) = tr ( A ) . {\displaystyle \operatorname {tr} \left(\mathbf {P} ^{-1}(\mathbf {A} \mathbf {P} )\right)=\operatorname {tr} \left((\mathbf {A} \mathbf {P} )\mathbf {P} ^{-1}\right)=\operatorname {tr} (\mathbf {A} ).} Similarity invariance is the crucial property of the trace in order to discuss traces of <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformations</a> as below. </p><p>Additionally, for real column vectors a ∈ R n {\displaystyle \mathbf {a} \in \mathbb {R} ^{n}} and b ∈ R n {\displaystyle \mathbf {b} \in \mathbb {R} ^{n}} , the trace of the outer product is equivalent to the inner product: </p> <p> tr ( b a T ) = a T b {\displaystyle \operatorname {tr} \left(\mathbf {b} \mathbf {a} ^{\textsf {T}}\right)=\mathbf {a} ^{\textsf {T}}\mathbf {b} } </p> <h3>Cyclic property</h3> <p>More generally, the trace is <i>invariant under <a href="/facts/Circular_shift/noSNJs87">circular shifts</a></i>, that is, </p> <p> tr ( A B C D ) = tr ( B C D A ) = tr ( C D A B ) = tr ( D A B C ) . {\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} \mathbf {D} )=\operatorname {tr} (\mathbf {B} \mathbf {C} \mathbf {D} \mathbf {A} )=\operatorname {tr} (\mathbf {C} \mathbf {D} \mathbf {A} \mathbf {B} )=\operatorname {tr} (\mathbf {D} \mathbf {A} \mathbf {B} \mathbf {C} ).} </p> <p>This is known as the <i>cyclic property</i>. </p><p>Arbitrary permutations are not allowed: in general, tr ( A B C ) ≠ tr ( A C B ) . {\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )\neq \operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} ).} </p><p>However, if products of <i>three</i> <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric</a> matrices are considered, any permutation is allowed, since: tr ( A B C ) = tr ( ( A B C ) T ) = tr ( C B A ) = tr ( A C B ) , {\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )=\operatorname {tr} \left(\left(\mathbf {A} \mathbf {B} \mathbf {C} \right)^{\mathsf {T}}\right)=\operatorname {tr} (\mathbf {C} \mathbf {B} \mathbf {A} )=\operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} ),} where the first equality is because the traces of a matrix and its transpose are equal. Note that this is not true in general for more than three factors. </p> <h3>Trace of a Kronecker product</h3> <p>The trace of the <a href="/facts/Kronecker_product/gABGx6I6">Kronecker product</a> of two matrices is the product of their traces: tr ( A ⊗ B ) = tr ( A ) tr ( B ) . {\displaystyle \operatorname {tr} (\mathbf {A} \otimes \mathbf {B} )=\operatorname {tr} (\mathbf {A} )\operatorname {tr} (\mathbf {B} ).} </p> <h3>Characterization of the trace</h3> <p>The following three properties: tr ( A + B ) = tr ( A ) + tr ( B ) , tr ( c A ) = c tr ( A ) , tr ( A B ) = tr ( B A ) , {\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} ),\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} ),\\\operatorname {tr} (\mathbf {A} \mathbf {B} )&=\operatorname {tr} (\mathbf {B} \mathbf {A} ),\end{aligned}}} characterize the trace <a href="/facts/Up_to/ydz4ztzX">up to</a> a scalar multiple in the following sense: If f {\displaystyle f} is a <a href="/facts/Linear_functional/oGh7Asrz">linear functional</a> on the space of square matrices that satisfies f ( x y ) = f ( y x ) , {\displaystyle f(xy)=f(yx),} then f {\displaystyle f} and tr {\displaystyle \operatorname {tr} } are proportional.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p><p>For n × n {\displaystyle n\times n} matrices, imposing the normalization f ( I ) = n {\displaystyle f(\mathbf {I} )=n} makes f {\displaystyle f} equal to the trace. </p> <h3>Trace as the sum of eigenvalues</h3> <p>Given any <i>n</i> × <i>n</i> matrix A, there is </p> <p> tr ( A ) = ∑ i = 1 n λ i {\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{n}\lambda _{i}} </p> <p>where λ1, ..., λ<i>n</i> are the <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> of A counted with multiplicity. This holds true even if A is a real matrix and some (or all) of the eigenvalues are complex numbers. This may be regarded as a consequence of the existence of the <a href="/facts/Jordan_canonical_form/M7ezYbnl">Jordan canonical form</a>, together with the similarity-invariance of the trace discussed above. </p> <h3>Trace of commutator</h3> <p>When both A and B are <i>n</i> × <i>n</i> matrices, the trace of the (ring-theoretic) <a href="/facts/Commutator/eYxzFmFY">commutator</a> of A and B vanishes: tr([A, B]) = 0, because tr(AB) = tr(BA) and tr is linear. One can state this as "the trace is a map of <a href="/facts/Lie_algebras/n2gAUkNq">Lie algebras</a> gl<i>n</i> → <i>k</i> from operators to scalars", as the commutator of scalars is trivial (it is an <a href="/facts/Abelian_Lie_algebra/n2gAUkNq">Abelian Lie algebra</a>). In particular, using similarity invariance, it follows that the identity matrix is never similar to the commutator of any pair of matrices. </p><p>Conversely, any square matrix with zero trace is a linear combination of the commutators of pairs of matrices.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> Moreover, any square matrix with zero trace is <a href="/facts/Bounded_operator/LYmDTIqR">unitarily equivalent</a> to a square matrix with diagonal consisting of all zeros. </p> <h3>Traces of special kinds of matrices</h3> <ul><li>The trace of the <i>n</i> × <i>n</i> <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a> is the dimension of the space, namely n. <p> tr ( I n ) = n {\displaystyle \operatorname {tr} \left(\mathbf {I} _{n}\right)=n} </p> This leads to <a href="/facts/Dimension_(vector_space)/z8m02A3W">generalizations of dimension using trace</a>.</li><li>The trace of a <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrix</a> is real, because the elements on the diagonal are real.</li><li>The trace of a <a href="/facts/Permutation_matrix/xC1ran5s">permutation matrix</a> is the number of <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed points</a> of the corresponding permutation, because the diagonal term <i>a</i><i>ii</i> is 1 if the <i>i</i>th point is fixed and 0 otherwise.</li><li>The trace of a <a href="/facts/Projection_(linear_algebra)/sElXGkxD">projection matrix</a> is the dimension of the target space. P X = X ( X T X ) − 1 X T ⟹ tr ( P X ) = rank ( X ) . {\displaystyle {\begin{aligned}\mathbf {P} _{\mathbf {X} }&=\mathbf {X} \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}\\[3pt]\Longrightarrow \operatorname {tr} \left(\mathbf {P} _{\mathbf {X} }\right)&=\operatorname {rank} (\mathbf {X} ).\end{aligned}}} The matrix PX is idempotent.</li><li>More generally, the trace of any <a href="/facts/Idempotent_matrix/88Es7cEn">idempotent matrix</a>, i.e. one with A2 = A, equals its own <a href="/facts/Rank_(linear_algebra)/Yp9b5LK3">rank</a>.</li><li>The trace of a <a href="/facts/Nilpotent_matrix/qIns3FJJ">nilpotent matrix</a> is zero. <p>When the characteristic of the base field is zero, the converse also holds: if tr(A<i>k</i>) = 0 for all k, then A is nilpotent. </p> When the characteristic <i>n</i> > 0 is positive, the identity in n dimensions is a counterexample, as tr ( I n k ) = tr ( I n ) = n ≡ 0 {\displaystyle \operatorname {tr} \left(\mathbf {I} _{n}^{k}\right)=\operatorname {tr} \left(\mathbf {I} _{n}\right)=n\equiv 0} , but the identity is not nilpotent.</li></ul> <h3>Relationship to the characteristic polynomial</h3> <p>The trace of an n × n {\displaystyle n\times n} matrix A {\displaystyle A} is the coefficient of t n − 1 {\displaystyle t^{n-1}} in the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a>, possibly changed of sign, according to the convention in the definition of the characteristic polynomial. </p> <h2 id="relationship-to-eigenvalues">Relationship to eigenvalues</h2> <p>If A is a linear operator represented by a square matrix with <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex</a> entries and if <i>λ</i>1, ..., <i>λn</i> are the <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> of A (listed according to their <a href="/facts/Algebraic_multiplicity/8TjEoT8u">algebraic multiplicities</a>), then </p> <p> tr ( A ) = ∑ i λ i {\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i}\lambda _{i}} </p> <p>This follows from the fact that A is always <a href="/facts/Similar_matrix/ySevpav1">similar</a> to its <a href="/facts/Jordan_form/M7ezYbnl">Jordan form</a>, an upper <a href="/facts/Triangular_matrix/qjXSFMM3">triangular matrix</a> having <i>λ</i>1, ..., <i>λn</i> on the main diagonal. In contrast, the <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of A is the <i>product</i> of its eigenvalues; that is, det ( A ) = ∏ i λ i . {\displaystyle \det(\mathbf {A} )=\prod _{i}\lambda _{i}.} </p><p>Everything in the present section applies as well to any square matrix with coefficients in an <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed field</a>. </p> <h3>Derivative relationships</h3> <p>If ΔA is a square matrix with small entries and I denotes the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>, then we have approximately </p><p> det ( I + Δ A ) ≈ 1 + tr ( Δ A ) . {\displaystyle \det(\mathbf {I} +\mathbf {\Delta A} )\approx 1+\operatorname {tr} (\mathbf {\Delta A} ).} </p><p>Precisely this means that the trace is the <a href="/facts/Derivative/7Ito70Dh">derivative</a> of the <a href="/facts/Determinant/eGYqJ2TF">determinant</a> function at the identity matrix. <a href="/facts/Jacobi%27s_formula/RhxPLByr">Jacobi's formula</a> </p><p> d det ( A ) = tr ( adj ( A ) ⋅ d A ) {\displaystyle d\det(\mathbf {A} )=\operatorname {tr} {\big (}\operatorname {adj} (\mathbf {A} )\cdot d\mathbf {A} {\big )}} </p><p>is more general and describes the <a href="/facts/Differential_(infinitesimal)/3dxU2Wy2">differential</a> of the determinant at an arbitrary square matrix, in terms of the trace and the <a href="/facts/Adjugate_matrix/jjT9v8ox">adjugate</a> of the matrix. </p><p>From this (or from the connection between the trace and the eigenvalues), one can derive a relation between the trace function, the <a href="/facts/Matrix_exponential/WID5cG2g">matrix exponential</a> function, and the determinant: det ( exp ( A ) ) = exp ( tr ( A ) ) . {\displaystyle \det(\exp(\mathbf {A} ))=\exp(\operatorname {tr} (\mathbf {A} )).} </p><p>A related characterization of the trace applies to linear <a href="/facts/Vector_field/ccFNuPPp">vector fields</a>. Given a matrix A, define a vector field F on R<i>n</i> by F(x) = Ax. The components of this vector field are linear functions (given by the rows of A). Its <a href="/facts/Divergence/zpwwkFoU">divergence</a> div F is a constant function, whose value is equal to tr(A). </p><p>By the <a href="/facts/Divergence_theorem/pUGKF1N2">divergence theorem</a>, one can interpret this in terms of flows: if F(x) represents the velocity of a fluid at location x and U is a region in R<i>n</i>, the <a href="/facts/Flow_network/k0LCoRwD">net flow</a> of the fluid out of U is given by tr(A) · vol(<i>U</i>), where vol(<i>U</i>) is the <a href="/facts/Volume/aeAXCBUd">volume</a> of U. </p><p>The trace is a linear operator, hence it commutes with the derivative: d tr ( X ) = tr ( d X ) . {\displaystyle d\operatorname {tr} (\mathbf {X} )=\operatorname {tr} (d\mathbf {X} ).} </p> <h2 id="trace-of-a-linear-operator">Trace of a linear operator</h2> <p>In general, given some linear map <i>f</i> : <i>V</i> → <i>V</i> (where V is a finite-<a href="/facts/Dimension_(linear_algebra)/z8m02A3W">dimensional</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a>), we can define the trace of this map by considering the trace of a <a href="/facts/Representation_theory/3ydLtaGH">matrix representation</a> of f, that is, choosing a <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> for V and describing f as a matrix relative to this basis, and taking the trace of this square matrix. The result will not depend on the basis chosen, since different bases will give rise to <a href="/facts/Matrix_similarity/ySevpav1">similar matrices</a>, allowing for the possibility of a basis-independent definition for the trace of a linear map. </p><p>Such a definition can be given using the <a href="/facts/Natural_isomorphism/xwPumTHt">canonical isomorphism</a> between the space End(<i>V</i>) of linear maps on V and <i>V</i> ⊗ <i>V</i>*, where <i>V</i>* is the <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> of V. Let v be in V and let g be in <i>V</i>*. Then the trace of the indecomposable element <i>v</i> ⊗ <i>g</i> is defined to be <i>g</i>(<i>v</i>); the trace of a general element is defined by linearity. The trace of a linear map <i>f</i> : <i>V</i> → <i>V</i> can then be defined as the trace, in the above sense, of the element of <i>V</i> ⊗ <i>V</i>* corresponding to <i>f</i> under the above mentioned canonical isomorphism. Using an explicit basis for V and the corresponding dual basis for <i>V</i>*, one can show that this gives the same definition of the trace as given above. </p> <h2 id="numerical-algorithms">Numerical algorithms</h2> <h3>Stochastic estimator</h3><p> The trace can be estimated unbiasedly by "Hutchinson's trick":<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a></p><p>Given any matrix W ∈ R n × n {\displaystyle {\boldsymbol {W}}\in \mathbb {R} ^{n\times n}} , and any random u ∈ R n {\displaystyle {\boldsymbol {u}}\in \mathbb {R} ^{n}} with E [ u u ⊺ ] = I {\displaystyle \mathbb {E} [{\boldsymbol {u}}{\boldsymbol {u}}^{\intercal }]=\mathbf {I} } , we have E [ u ⊺ W u ] = tr W {\displaystyle \mathbb {E} [{\boldsymbol {u}}^{\intercal }{\boldsymbol {W}}{\boldsymbol {u}}]=\operatorname {tr} {\boldsymbol {W}}} . </p> <p>For a proof expand the expectation directly. </p><p>Usually, the random vector is sampled from N ( 0 , I ) {\displaystyle \operatorname {N} (\mathbf {0} ,\mathbf {I} )} (normal distribution) or { ± n − 1 / 2 } n {\displaystyle \{\pm n^{-1/2}\}^{n}} (<a href="/facts/Rademacher_distribution/MlPxl57G">Rademacher distribution</a>). </p><p>More sophisticated stochastic estimators of trace have been developed.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> </p> <h2 id="applications">Applications</h2> <p>If a 2 x 2 real matrix has zero trace, its square is a <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a>. </p><p>The trace of a 2 × 2 <a href="/facts/Complex_matrix/qa8FI4ko">complex matrix</a> is used to classify <a href="/facts/M%C3%B6bius_transformation/YEkTTLfK">Möbius transformations</a>. First, the matrix is normalized to make its <a href="/facts/Determinant/eGYqJ2TF">determinant</a> equal to one. Then, if the square of the trace is 4, the corresponding transformation is <i>parabolic</i>. If the square is in the interval [0,4), it is <i>elliptic</i>. Finally, if the square is greater than 4, the transformation is <i>loxodromic</i>. See <a href="/facts/M%C3%B6bius_transformation/YEkTTLfK">classification of Möbius transformations</a>. </p><p>The trace is used to define <a href="/facts/Character_(mathematics)/M46SyYUp">characters</a> of <a href="/facts/Group_representation/FsuuDagQ">group representations</a>. Two representations A, B : <i>G</i> → <i>GL</i>(<i>V</i>) of a group G are equivalent (up to change of basis on V) if tr(A(<i>g</i>)) = tr(B(<i>g</i>)) for all <i>g</i> ∈ <i>G</i>. </p><p>The trace also plays a central role in the distribution of <a href="/facts/Quadratic_form_(statistics)/qH3sK2VG">quadratic forms</a>. </p> <h2 id="lie-algebra">Lie algebra</h2> <p>The trace is a map of Lie algebras tr : g l n → K {\displaystyle \operatorname {tr} :{\mathfrak {gl}}_{n}\to K} from the Lie algebra g l n {\displaystyle {\mathfrak {gl}}_{n}} of linear operators on an n-dimensional space (<i>n</i> × <i>n</i> matrices with entries in K {\displaystyle K} ) to the Lie algebra K of scalars; as K is Abelian (the Lie bracket vanishes), the fact that this is a map of Lie algebras is exactly the statement that the trace of a bracket vanishes: tr ( [ A , B ] ) = 0 for each A , B ∈ g l n . {\displaystyle \operatorname {tr} ([\mathbf {A} ,\mathbf {B} ])=0{\text{ for each }}\mathbf {A} ,\mathbf {B} \in {\mathfrak {gl}}_{n}.} </p><p>The kernel of this map, a matrix whose trace is <a href="/facts/0_(number)/4NBRtKHc">zero</a>, is often said to be traceless or trace free, and these matrices form the <a href="/facts/Simple_Lie_algebra/Hzmh6rYT">simple Lie algebra</a> s l n {\displaystyle {\mathfrak {sl}}_{n}} , which is the <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> of the <a href="/facts/Special_linear_group/rsahah9w">special linear group</a> of matrices with determinant 1. The special linear group consists of the matrices which do not change volume, while the <a href="/facts/Special_linear_Lie_algebra/IpHvcM2t">special linear Lie algebra</a> is the matrices which do not alter volume of <i>infinitesimal</i> sets. </p><p>In fact, there is an internal <a href="/facts/Direct_sum_of_Lie_algebras/Y6PBJnjg">direct sum</a> decomposition g l n = s l n ⊕ K {\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K} of operators/matrices into traceless operators/matrices and scalars operators/matrices. The projection map onto scalar operators can be expressed in terms of the trace, concretely as: A ↦ 1 n tr ( A ) I . {\displaystyle \mathbf {A} \mapsto {\frac {1}{n}}\operatorname {tr} (\mathbf {A} )\mathbf {I} .} </p><p>Formally, one can compose the trace (the <a href="/facts/Counit/pvx52xS6">counit</a> map) with the unit map K → g l n {\displaystyle K\to {\mathfrak {gl}}_{n}} of "inclusion of <a href="/facts/Scalar_transformation/tjIAHrE6">scalars</a>" to obtain a map g l n → g l n {\displaystyle {\mathfrak {gl}}_{n}\to {\mathfrak {gl}}_{n}} mapping onto scalars, and multiplying by n. Dividing by n makes this a projection, yielding the formula above. </p><p>In terms of <a href="/facts/Short_exact_sequence/leUJnwJb">short exact sequences</a>, one has 0 → s l n → g l n → tr K → 0 {\displaystyle 0\to {\mathfrak {sl}}_{n}\to {\mathfrak {gl}}_{n}{\overset {\operatorname {tr} }{\to }}K\to 0} which is analogous to 1 → SL n → GL n → det K ∗ → 1 {\displaystyle 1\to \operatorname {SL} _{n}\to \operatorname {GL} _{n}{\overset {\det }{\to }}K^{*}\to 1} (where K ∗ = K ∖ { 0 } {\displaystyle K^{*}=K\setminus \{0\}} ) for <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a>. However, the trace splits naturally (via 1 / n {\displaystyle 1/n} times scalars) so g l n = s l n ⊕ K {\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K} , but the splitting of the determinant would be as the nth root times scalars, and this does not in general define a function, so the determinant does not split and the general linear group does not decompose: GL n ≠ SL n × K ∗ . {\displaystyle \operatorname {GL} _{n}\neq \operatorname {SL} _{n}\times K^{*}.} </p> <h3>Bilinear forms</h3> <p>The <a href="/facts/Bilinear_form/gyvzn9ym">bilinear form</a> (where X, Y are square matrices) B ( X , Y ) = tr ( ad ( X ) ad ( Y ) ) where ad ( X ) Y = [ X , Y ] = X Y − Y X {\displaystyle B(\mathbf {X} ,\mathbf {Y} )=\operatorname {tr} (\operatorname {ad} (\mathbf {X} )\operatorname {ad} (\mathbf {Y} ))\quad {\text{where }}\operatorname {ad} (\mathbf {X} )\mathbf {Y} =[\mathbf {X} ,\mathbf {Y} ]=\mathbf {X} \mathbf {Y} -\mathbf {Y} \mathbf {X} } is called the <a href="/facts/Killing_form/u4HxRh5o">Killing form</a>, which is used for the classification of Lie algebras. </p><p>The trace defines a bilinear form: ( X , Y ) ↦ tr ( X Y ) . {\displaystyle (\mathbf {X} ,\mathbf {Y} )\mapsto \operatorname {tr} (\mathbf {X} \mathbf {Y} ).} </p><p>The form is symmetric, non-degenerate<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> and associative in the sense that: tr ( X [ Y , Z ] ) = tr ( [ X , Y ] Z ) . {\displaystyle \operatorname {tr} (\mathbf {X} [\mathbf {Y} ,\mathbf {Z} ])=\operatorname {tr} ([\mathbf {X} ,\mathbf {Y} ]\mathbf {Z} ).} </p><p>For a complex simple Lie algebra (such as s l {\displaystyle {\mathfrak {sl}}} <i>n</i>), every such bilinear form is proportional to each other; in particular, to the Killing form. </p><p>Two matrices X and Y are said to be <i>trace orthogonal</i> if tr ( X Y ) = 0. {\displaystyle \operatorname {tr} (\mathbf {X} \mathbf {Y} )=0.} </p><p>There is a generalization to a general representation ( ρ , g , V ) {\displaystyle (\rho ,{\mathfrak {g}},V)} of a Lie algebra g {\displaystyle {\mathfrak {g}}} , such that ρ {\displaystyle \rho } is a homomorphism of Lie algebras ρ : g → End ( V ) . {\displaystyle \rho :{\mathfrak {g}}\rightarrow {\text{End}}(V).} The trace form tr V {\displaystyle {\text{tr}}_{V}} on End ( V ) {\displaystyle {\text{End}}(V)} is defined as above. The bilinear form ϕ ( X , Y ) = tr V ( ρ ( X ) ρ ( Y ) ) {\displaystyle \phi (\mathbf {X} ,\mathbf {Y} )={\text{tr}}_{V}(\rho (\mathbf {X} )\rho (\mathbf {Y} ))} is symmetric and invariant due to cyclicity. </p> <h2 id="generalizations">Generalizations</h2> <p>The concept of trace of a matrix is generalized to the <a href="/facts/Trace_class/2AmWENA3">trace class</a> of <a href="/facts/Compact_operator/KNkcK4mE">compact operators</a> on <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a>, and the analog of the <a href="/facts/Frobenius_norm/UJvsb9fF">Frobenius norm</a> is called the <a href="/facts/Hilbert%E2%80%93Schmidt_operator/Xhy17yn6">Hilbert–Schmidt</a> norm. </p><p>If K is a trace-class operator, then for any <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a> ( e n ) n {\displaystyle (e_{n})_{n}} , the trace is given by tr ( K ) = ∑ n ⟨ e n , K e n ⟩ , {\displaystyle \operatorname {tr} (K)=\sum _{n}\left\langle e_{n},Ke_{n}\right\rangle ,} and is finite and independent of the orthonormal basis.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> </p><p>The <a href="/facts/Partial_trace/ra7RhkkT">partial trace</a> is another generalization of the trace that is operator-valued. The trace of a linear operator Z which lives on a product space <i>A</i> ⊗ <i>B</i> is equal to the partial traces over A and B: tr ( Z ) = tr A ( tr B ( Z ) ) = tr B ( tr A ( Z ) ) . {\displaystyle \operatorname {tr} (Z)=\operatorname {tr} _{A}\left(\operatorname {tr} _{B}(Z)\right)=\operatorname {tr} _{B}\left(\operatorname {tr} _{A}(Z)\right).} </p><p>For more properties and a generalization of the partial trace, see <a href="/facts/Traced_monoidal_category/Wjwh6mEe">traced monoidal categories</a>. </p><p>If A is a general <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> over a field k, then a trace on A is often defined to be any map tr : <i>A</i> ↦ <i>k</i> which vanishes on commutators; tr([<i>a</i>,<i>b</i>]) = 0 for all <i>a</i>, <i>b</i> ∈ <i>A</i>. Such a trace is not uniquely defined; it can always at least be modified by multiplication by a nonzero scalar. </p><p>A <a href="/facts/Supertrace/9IC2DHwu">supertrace</a> is the generalization of a trace to the setting of <a href="/facts/Superalgebra/96vSHP12">superalgebras</a>. </p><p>The operation of <a href="/facts/Tensor_contraction/fqz0zvWp">tensor contraction</a> generalizes the trace to arbitrary tensors. </p><p>Gomme and Klein (2011) define a matrix trace operator trm {\displaystyle \operatorname {trm} } that operates on <a href="/facts/Block_matrix/fPI8HX9f">block matrices</a> and use it to compute second-order perturbation solutions to dynamic economic models without the need for <a href="/facts/Tensor_notation/QL2GnCmr">tensor notation</a>.<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> </p> <h2 id="traces-in-the-language-of-tensor-products">Traces in the language of tensor products</h2> <p>Given a vector space V, there is a natural bilinear map <i>V</i> × <i>V</i>∗ → <i>F</i> given by sending (<i>v</i>, φ) to the scalar φ(<i>v</i>). The <a href="/facts/Tensor_product/7K3pR1ap">universal property</a> of the <a href="/facts/Tensor_product/7K3pR1ap">tensor product</a> <i>V</i> ⊗ <i>V</i>∗ automatically implies that this bilinear map is induced by a linear functional on <i>V</i> ⊗ <i>V</i>∗.<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> </p><p>Similarly, there is a natural bilinear map <i>V</i> × <i>V</i>∗ → Hom(<i>V</i>, <i>V</i>) given by sending (<i>v</i>, φ) to the linear map <i>w</i> ↦ φ(<i>w</i>)<i>v</i>. The universal property of the tensor product, just as used previously, says that this bilinear map is induced by a linear map <i>V</i> ⊗ <i>V</i>∗ → Hom(<i>V</i>, <i>V</i>). If V is finite-dimensional, then this linear map is a <a href="/facts/Linear_isomorphism/Q1PBKNVV">linear isomorphism</a>.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> This fundamental fact is a straightforward consequence of the existence of a (finite) basis of V, and can also be phrased as saying that any linear map <i>V</i> → <i>V</i> can be written as the sum of (finitely many) rank-one linear maps. Composing the inverse of the isomorphism with the linear functional obtained above results in a linear functional on Hom(<i>V</i>, <i>V</i>). This linear functional is exactly the same as the trace. </p><p>Using the definition of trace as the sum of diagonal elements, the matrix formula tr(AB) = tr(BA) is straightforward to prove, and was given above. In the present perspective, one is considering linear maps S and T, and viewing them as sums of rank-one maps, so that there are linear functionals <i>φ</i><i>i</i> and <i>ψ</i><i>j</i> and nonzero vectors <i>v</i><i>i</i> and <i>w</i><i>j</i> such that <i>S</i>(u) = Σ<i>φ</i><i>i</i>(<i>u</i>)<i>v</i><i>i</i> and <i>T</i>(u) = Σ<i>ψ</i><i>j</i>(<i>u</i>)<i>w</i><i>j</i> for any u in V. Then </p> ( S ∘ T ) ( u ) = ∑ i φ i ( ∑ j ψ j ( u ) w j ) v i = ∑ i ∑ j ψ j ( u ) φ i ( w j ) v i {\displaystyle (S\circ T)(u)=\sum _{i}\varphi _{i}\left(\sum _{j}\psi _{j}(u)w_{j}\right)v_{i}=\sum _{i}\sum _{j}\psi _{j}(u)\varphi _{i}(w_{j})v_{i}} <p>for any u in V. The rank-one linear map <i>u</i> ↦ <i>ψ</i><i>j</i>(<i>u</i>)<i>φ</i><i>i</i>(<i>w</i><i>j</i>)<i>v</i><i>i</i> has trace <i>ψ</i><i>j</i>(<i>v</i><i>i</i>)<i>φ</i><i>i</i>(<i>w</i><i>j</i>) and so </p> tr ( S ∘ T ) = ∑ i ∑ j ψ j ( v i ) φ i ( w j ) = ∑ j ∑ i φ i ( w j ) ψ j ( v i ) . {\displaystyle \operatorname {tr} (S\circ T)=\sum _{i}\sum _{j}\psi _{j}(v_{i})\varphi _{i}(w_{j})=\sum _{j}\sum _{i}\varphi _{i}(w_{j})\psi _{j}(v_{i}).} <p>Following the same procedure with S and T reversed, one finds exactly the same formula, proving that tr(<i>S</i> ∘ <i>T</i>) equals tr(<i>T</i> ∘ <i>S</i>). </p><p>The above proof can be regarded as being based upon tensor products, given that the fundamental identity of End(<i>V</i>) with <i>V</i> ⊗ <i>V</i>∗ is equivalent to the expressibility of any linear map as the sum of rank-one linear maps. As such, the proof may be written in the notation of tensor products. Then one may consider the multilinear map <i>V</i> × <i>V</i>∗ × <i>V</i> × <i>V</i>∗ → <i>V</i> ⊗ <i>V</i>∗ given by sending (<i>v</i>, <i>φ</i>, <i>w</i>, <i>ψ</i>) to <i>φ</i>(<i>w</i>)<i>v</i> ⊗ <i>ψ</i>. Further composition with the trace map then results in <i>φ</i>(<i>w</i>)<i>ψ</i>(<i>v</i>), and this is unchanged if one were to have started with (<i>w</i>, <i>ψ</i>, <i>v</i>, <i>φ</i>) instead. One may also consider the bilinear map End(<i>V</i>) × End(<i>V</i>) → End(<i>V</i>) given by sending (<i>f</i>, <i>g</i>) to the composition <i>f</i> ∘ <i>g</i>, which is then induced by a linear map End(<i>V</i>) ⊗ End(<i>V</i>) → End(<i>V</i>). It can be seen that this coincides with the linear map <i>V</i> ⊗ <i>V</i>∗ ⊗ <i>V</i> ⊗ <i>V</i>∗ → <i>V</i> ⊗ <i>V</i>∗. The established symmetry upon composition with the trace map then establishes the equality of the two traces.<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a> </p><p>For any finite dimensional vector space V, there is a natural linear map <i>F</i> → <i>V</i> ⊗ <i>V</i>'; in the language of linear maps, it assigns to a scalar c the linear map <i>c</i>⋅id<i>V</i>. Sometimes this is called <i>coevaluation map</i>, and the trace <i>V</i> ⊗ <i>V</i>' → <i>F</i> is called <i>evaluation map</i>.<a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> These structures can be axiomatized to define <a href="/facts/Categorical_trace/cf731ABA">categorical traces</a> in the abstract setting of <a href="/facts/Category_theory/6zS3DXPa">category theory</a>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Scalar_curvature/tfgm3AuK">Trace of a tensor with respect to a metric tensor</a></li> <li><a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">Characteristic function</a></li> <li><a href="/facts/Field_trace/shd5K3gR">Field trace</a></li> <li><a href="/facts/Golden%E2%80%93Thompson_inequality/ys4eXYV5">Golden–Thompson inequality</a></li> <li><a href="/facts/Singular_trace/S4xkdjCI">Singular trace</a></li> <li><a href="/facts/Specht%27s_theorem/GflRK0el">Specht's theorem</a></li> <li><a href="/facts/Trace_class/2AmWENA3">Trace class</a></li> <li><a href="/facts/Trace_identity/nSC0fdLP">Trace identity</a></li> <li><a href="/facts/Trace_inequalities/jtYypUYj">Trace inequalities</a></li> <li><a href="/facts/Von_Neumann%27s_trace_inequality/jtYypUYj">von Neumann's trace inequality</a></li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Felix_Gantmacher/cWd0KU9A">Gantmacher, F.R.</a> (1959). <i>The Theory of Matrices</i>. Translated by <a href="/facts/Kurt_Hirsch/YmdGzYaQ">Hirsch, K.A.</a> New York, NY: <a href="/facts/Chelsea_Publishing_Company/KWAyXCyF">Chelsea Publishing Company</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0107649">0107649</a>.</li></ul> <ul><li><a href="/facts/Roger_Horn/w1J0ArVI">Horn, R.A.</a>; <a href="/facts/Charles_Royal_Johnson/Qehgl6tm">Johnson, C.R.</a> (2013) [1985]. <i>Matrix Analysis</i> (2nd ed.). Cambridge, UK: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-54823-6. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2978290">2978290</a>.</li></ul> <ul><li><a href="/facts/Gilbert_Strang/bbzaoPsm">Strang, G.</a> (2004) [1976]. <i>Linear Algebra and its Applications</i> (4th ed.). <a href="/facts/Cengage_Learning/Bda6bYXT">Cengage Learning</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-003010567-8.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Trace_of_a_square_matrix">"Trace of a square matrix"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Rank, trace, determinant, transpose, and inverse of matrices". fourier.eng.hmc.edu. Retrieved 2020-09-09. <a href="http://fourier.eng.hmc.edu/e161/lectures/algebra/node2.html" target="_blank">http://fourier.eng.hmc.edu/e161/lectures/algebra/node2.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"Rank, trace, determinant, transpose, and inverse of matrices". fourier.eng.hmc.edu. Retrieved 2020-09-09. <a href="http://fourier.eng.hmc.edu/e161/lectures/algebra/node2.html" target="_blank">http://fourier.eng.hmc.edu/e161/lectures/algebra/node2.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Weisstein, Eric W. (2003) [1999]. "Trace (matrix)". In Weisstein, Eric W. (ed.). CRC Concise Encyclopedia of Mathematics (2nd ed.). Boca Raton, FL: Chapman & Hall. doi:10.1201/9781420035223. ISBN 1-58488-347-2. MR 1944431. Zbl 1079.00009. Retrieved 2020-09-09. <a href="1-58488-347-2" target="_blank">1-58488-347-2</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Lipschutz, Seymour; Lipson, Marc (September 2005). Theory and Problems of Linear Algebra. Schaum's Outline. McGraw-Hill. ISBN 9780070605022. <a href="9780070605022" target="_blank">9780070605022</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"Rank, trace, determinant, transpose, and inverse of matrices". fourier.eng.hmc.edu. Retrieved 2020-09-09. <a href="http://fourier.eng.hmc.edu/e161/lectures/algebra/node2.html" target="_blank">http://fourier.eng.hmc.edu/e161/lectures/algebra/node2.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Weisstein, Eric W. (2003) [1999]. "Trace (matrix)". In Weisstein, Eric W. (ed.). CRC Concise Encyclopedia of Mathematics (2nd ed.). Boca Raton, FL: Chapman & Hall. doi:10.1201/9781420035223. ISBN 1-58488-347-2. MR 1944431. Zbl 1079.00009. Retrieved 2020-09-09. <a href="1-58488-347-2" target="_blank">1-58488-347-2</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Lipschutz, Seymour; Lipson, Marc (September 2005). Theory and Problems of Linear Algebra. Schaum's Outline. McGraw-Hill. ISBN 9780070605022. <a href="9780070605022" target="_blank">9780070605022</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>"Rank, trace, determinant, transpose, and inverse of matrices". fourier.eng.hmc.edu. Retrieved 2020-09-09. <a href="http://fourier.eng.hmc.edu/e161/lectures/algebra/node2.html" target="_blank">http://fourier.eng.hmc.edu/e161/lectures/algebra/node2.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Weisstein, Eric W. (2003) [1999]. "Trace (matrix)". In Weisstein, Eric W. (ed.). CRC Concise Encyclopedia of Mathematics (2nd ed.). Boca Raton, FL: Chapman & Hall. doi:10.1201/9781420035223. ISBN 1-58488-347-2. MR 1944431. Zbl 1079.00009. Retrieved 2020-09-09. <a href="1-58488-347-2" target="_blank">1-58488-347-2</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Lipschutz, Seymour; Lipson, Marc (September 2005). Theory and Problems of Linear Algebra. Schaum's Outline. McGraw-Hill. ISBN 9780070605022. <a href="9780070605022" target="_blank">9780070605022</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis (2nd ed.). Cambridge University Press. ISBN 9780521839402. <a href="9780521839402" target="_blank">9780521839402</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>"Rank, trace, determinant, transpose, and inverse of matrices". fourier.eng.hmc.edu. Retrieved 2020-09-09. <a href="http://fourier.eng.hmc.edu/e161/lectures/algebra/node2.html" target="_blank">http://fourier.eng.hmc.edu/e161/lectures/algebra/node2.html</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Weisstein, Eric W. (2003) [1999]. "Trace (matrix)". In Weisstein, Eric W. (ed.). CRC Concise Encyclopedia of Mathematics (2nd ed.). Boca Raton, FL: Chapman & Hall. doi:10.1201/9781420035223. ISBN 1-58488-347-2. MR 1944431. Zbl 1079.00009. Retrieved 2020-09-09. <a href="1-58488-347-2" target="_blank">1-58488-347-2</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Lipschutz, Seymour; Lipson, Marc (September 2005). Theory and Problems of Linear Algebra. Schaum's Outline. McGraw-Hill. ISBN 9780070605022. <a href="9780070605022" target="_blank">9780070605022</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>This is immediate from the definition of the matrix product: tr ( A B ) = ∑ i = 1 m ( A B ) i i = ∑ i = 1 m ∑ j = 1 n a i j b j i = ∑ j = 1 n ∑ i = 1 m b j i a i j = ∑ j = 1 n ( B A ) j j = tr ( B A ) . {\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} )=\sum _{i=1}^{m}\left(\mathbf {A} \mathbf {B} \right)_{ii}=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ji}=\sum _{j=1}^{n}\sum _{i=1}^{m}b_{ji}a_{ij}=\sum _{j=1}^{n}\left(\mathbf {B} \mathbf {A} \right)_{jj}=\operatorname {tr} (\mathbf {B} \mathbf {A} ).} <a href="/wiki/Matrix_product" target="_blank">/wiki/Matrix_product</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>For example, if A = ( 0 1 0 0 ) , B = ( 0 0 1 0 ) , {\displaystyle \mathbf {A} ={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad \mathbf {B} ={\begin{pmatrix}0&0\\1&0\end{pmatrix}},} then the product is A B = ( 1 0 0 0 ) , {\displaystyle \mathbf {AB} ={\begin{pmatrix}1&0\\0&0\end{pmatrix}},} and the traces are tr(AB) = 1 ≠ 0 ⋅ 0 = tr(A)tr(B). <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Proof: Let e i j {\displaystyle e_{ij}} the standard basis and note that f ( e i j ) = f ( e i e j ⊤ ) = f ( e i e 1 ⊤ e 1 e j ⊤ ) = f ( e 1 e j ⊤ e i e 1 ⊤ ) = f ( 0 ) = 0 {\displaystyle f\left(e_{ij}\right)=f\left(e_{i}e_{j}^{\top }\right)=f\left(e_{i}e_{1}^{\top }e_{1}e_{j}^{\top }\right)=f\left(e_{1}e_{j}^{\top }e_{i}e_{1}^{\top }\right)=f\left(0\right)=0} if i ≠ j {\displaystyle i\neq j} and f ( e j j ) = f ( e 11 ) {\displaystyle f\left(e_{jj}\right)=f\left(e_{11}\right)} f ( A ) = ∑ i , j [ A ] i j f ( e i j ) = ∑ i [ A ] i i f ( e 11 ) = f ( e 11 ) tr ( A ) . {\displaystyle f(\mathbf {A} )=\sum _{i,j}[\mathbf {A} ]_{ij}f\left(e_{ij}\right)=\sum _{i}[\mathbf {A} ]_{ii}f\left(e_{11}\right)=f\left(e_{11}\right)\operatorname {tr} (\mathbf {A} ).} More abstractly, this corresponds to the decomposition g l n = s l n ⊕ k , {\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus k,} as tr ( A B ) = tr ( B A ) {\displaystyle \operatorname {tr} (AB)=\operatorname {tr} (BA)} (equivalently, tr ( [ A , B ] ) = 0 {\displaystyle \operatorname {tr} ([A,B])=0} ) defines the trace on s l n , {\displaystyle {\mathfrak {sl}}_{n},} which has complement the scalar matrices, and leaves one degree of freedom: any such map is determined by its value on scalars, which is one scalar parameter and hence all are multiple of the trace, a nonzero such map. <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Proof: s l n {\displaystyle {\mathfrak {sl}}_{n}} is a semisimple Lie algebra and thus every element in it is a linear combination of commutators of some pairs of elements, otherwise the derived algebra would be a proper ideal. <a href="/wiki/Semisimple_Lie_algebra" target="_blank">/wiki/Semisimple_Lie_algebra</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Hutchinson, M.F. (January 1989). "A Stochastic Estimator of the Trace of the Influence Matrix for Laplacian Smoothing Splines". Communications in Statistics - Simulation and Computation. 18 (3): 1059–1076. doi:10.1080/03610918908812806. ISSN 0361-0918. <a href="http://www.tandfonline.com/doi/abs/10.1080/03610918908812806" target="_blank">http://www.tandfonline.com/doi/abs/10.1080/03610918908812806</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Avron, Haim; Toledo, Sivan (2011-04-11). "Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix". Journal of the ACM. 58 (2): 8:1–8:34. doi:10.1145/1944345.1944349. ISSN 0004-5411. S2CID 5827717. <a href="https://doi.org/10.1145/1944345.1944349" target="_blank">https://doi.org/10.1145/1944345.1944349</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>This follows from the fact that tr(A*A) = 0 if and only if A = 0. <a href="/wiki/If_and_only_if" target="_blank">/wiki/If_and_only_if</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Teschl, G. (30 October 2014). Mathematical Methods in Quantum Mechanics. Graduate Studies in Mathematics. Vol. 157 (2nd ed.). American Mathematical Society. ISBN 978-1470417048. <a href="978-1470417048" target="_blank">978-1470417048</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>P. Gomme, P. Klein (2011). "Second-order approximation of dynamic models without the use of tensors". Journal of Economic Dynamics & Control. 35 (4): 604–615. doi:10.1016/j.jedc.2010.10.006. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Kassel, Christian (1995). Quantum groups. Graduate Texts in Mathematics. Vol. 155. New York: Springer-Verlag. doi:10.1007/978-1-4612-0783-2. ISBN 0-387-94370-6. MR 1321145. Zbl 0808.17003. <a href="0-387-94370-6" target="_blank">0-387-94370-6</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>Kassel, Christian (1995). Quantum groups. Graduate Texts in Mathematics. Vol. 155. New York: Springer-Verlag. doi:10.1007/978-1-4612-0783-2. ISBN 0-387-94370-6. MR 1321145. Zbl 0808.17003. <a href="0-387-94370-6" target="_blank">0-387-94370-6</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>Kassel, Christian (1995). Quantum groups. Graduate Texts in Mathematics. Vol. 155. New York: Springer-Verlag. doi:10.1007/978-1-4612-0783-2. ISBN 0-387-94370-6. MR 1321145. Zbl 0808.17003. <a href="0-387-94370-6" target="_blank">0-387-94370-6</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>Kassel, Christian (1995). Quantum groups. Graduate Texts in Mathematics. Vol. 155. New York: Springer-Verlag. doi:10.1007/978-1-4612-0783-2. ISBN 0-387-94370-6. MR 1321145. Zbl 0808.17003. <a href="0-387-94370-6" target="_blank">0-387-94370-6</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> </ol>