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Nilpotent matrix
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<h2 id="examples">Examples</h2> <h3>Example 1</h3> <p>The matrix </p> A = [ 0 1 0 0 ] {\displaystyle A={\begin{bmatrix}0&1\\0&0\end{bmatrix}}} <p>is nilpotent with index 2, since A 2 = 0 {\displaystyle A^{2}=0} . </p> <h3>Example 2</h3> <p>More generally, any n {\displaystyle n} -dimensional <a href="/facts/Triangular_matrix/qjXSFMM3">triangular matrix</a> with zeros along the <a href="/facts/Main_diagonal/rkT2F08Q">main diagonal</a> is nilpotent, with index ≤ n {\displaystyle \leq n} . For example, the matrix </p> B = [ 0 2 1 6 0 0 1 2 0 0 0 3 0 0 0 0 ] {\displaystyle B={\begin{bmatrix}0&2&1&6\\0&0&1&2\\0&0&0&3\\0&0&0&0\end{bmatrix}}} <p>is nilpotent, with </p> B 2 = [ 0 0 2 7 0 0 0 3 0 0 0 0 0 0 0 0 ] ; B 3 = [ 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 ] ; B 4 = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] {\displaystyle B^{2}={\begin{bmatrix}0&0&2&7\\0&0&0&3\\0&0&0&0\\0&0&0&0\end{bmatrix}};\ B^{3}={\begin{bmatrix}0&0&0&6\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}};\ B^{4}={\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}}} <p>The index of B {\displaystyle B} is therefore 4. </p> <h3>Example 3</h3> <p>Although the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example, </p> C = [ 5 − 3 2 15 − 9 6 10 − 6 4 ] C 2 = [ 0 0 0 0 0 0 0 0 0 ] {\displaystyle C={\begin{bmatrix}5&-3&2\\15&-9&6\\10&-6&4\end{bmatrix}}\qquad C^{2}={\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}}} <p>although the matrix has no zero entries. </p> <h3>Example 4</h3> <p>Additionally, any matrices of the form </p> [ a 1 a 1 ⋯ a 1 a 2 a 2 ⋯ a 2 ⋮ ⋮ ⋱ ⋮ − a 1 − a 2 − … − a n − 1 − a 1 − a 2 − … − a n − 1 … − a 1 − a 2 − … − a n − 1 ] {\displaystyle {\begin{bmatrix}a_{1}&a_{1}&\cdots &a_{1}\\a_{2}&a_{2}&\cdots &a_{2}\\\vdots &\vdots &\ddots &\vdots \\-a_{1}-a_{2}-\ldots -a_{n-1}&-a_{1}-a_{2}-\ldots -a_{n-1}&\ldots &-a_{1}-a_{2}-\ldots -a_{n-1}\end{bmatrix}}} <p>such as </p> [ 5 5 5 6 6 6 − 11 − 11 − 11 ] {\displaystyle {\begin{bmatrix}5&5&5\\6&6&6\\-11&-11&-11\end{bmatrix}}} <p>or </p> [ 1 1 1 1 2 2 2 2 4 4 4 4 − 7 − 7 − 7 − 7 ] {\displaystyle {\begin{bmatrix}1&1&1&1\\2&2&2&2\\4&4&4&4\\-7&-7&-7&-7\end{bmatrix}}} <p>square to zero. </p> <h3>Example 5</h3> <p>Perhaps some of the most striking examples of nilpotent matrices are n × n {\displaystyle n\times n} square matrices of the form: </p> [ 2 2 2 ⋯ 1 − n n + 2 1 1 ⋯ − n 1 n + 2 1 ⋯ − n 1 1 n + 2 ⋯ − n ⋮ ⋮ ⋮ ⋱ ⋮ ] {\displaystyle {\begin{bmatrix}2&2&2&\cdots &1-n\\n+2&1&1&\cdots &-n\\1&n+2&1&\cdots &-n\\1&1&n+2&\cdots &-n\\\vdots &\vdots &\vdots &\ddots &\vdots \end{bmatrix}}} <p>The first few of which are: </p> [ 2 − 1 4 − 2 ] [ 2 2 − 2 5 1 − 3 1 5 − 3 ] [ 2 2 2 − 3 6 1 1 − 4 1 6 1 − 4 1 1 6 − 4 ] [ 2 2 2 2 − 4 7 1 1 1 − 5 1 7 1 1 − 5 1 1 7 1 − 5 1 1 1 7 − 5 ] … {\displaystyle {\begin{bmatrix}2&-1\\4&-2\end{bmatrix}}\qquad {\begin{bmatrix}2&2&-2\\5&1&-3\\1&5&-3\end{bmatrix}}\qquad {\begin{bmatrix}2&2&2&-3\\6&1&1&-4\\1&6&1&-4\\1&1&6&-4\end{bmatrix}}\qquad {\begin{bmatrix}2&2&2&2&-4\\7&1&1&1&-5\\1&7&1&1&-5\\1&1&7&1&-5\\1&1&1&7&-5\end{bmatrix}}\qquad \ldots } <p>These matrices are nilpotent but there are no zero entries in any powers of them less than the index.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> <h3>Example 6</h3> <p>Consider the linear space of <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> of a bounded degree. The <a href="/facts/Derivative/7Ito70Dh">derivative</a> operator is a linear map. We know that applying the derivative to a polynomial decreases its degree by one, so when applying it iteratively, we will eventually obtain zero. Therefore, on such a space, the derivative is representable by a nilpotent matrix. </p> <h2 id="characterization">Characterization</h2> <p>For an n × n {\displaystyle n\times n} square matrix N {\displaystyle N} with <a href="/facts/Real_number/R02gw5Pb">real</a> (or <a href="/facts/Complex_number/2w7mqI7e">complex</a>) entries, the following are equivalent: </p> <ul><li> N {\displaystyle N} is nilpotent.</li> <li>The <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> for N {\displaystyle N} is det ( x I − N ) = x n {\displaystyle \det \left(xI-N\right)=x^{n}} .</li> <li>The <a href="/facts/Minimal_polynomial_(linear_algebra)/kv3lQb64">minimal polynomial</a> for N {\displaystyle N} is x k {\displaystyle x^{k}} for some positive integer k ≤ n {\displaystyle k\leq n} .</li> <li>The only complex eigenvalue for N {\displaystyle N} is 0.</li></ul> <p>The last theorem holds true for matrices over any <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of characteristic 0 or sufficiently large characteristic. (cf. <a href="/facts/Newton%2527s_identities/5J9flXxA">Newton's identities</a>) </p><p>This theorem has several consequences, including: </p> <ul><li>The index of an n × n {\displaystyle n\times n} nilpotent matrix is always less than or equal to n {\displaystyle n} . For example, every 2 × 2 {\displaystyle 2\times 2} nilpotent matrix squares to zero.</li> <li>The <a href="/facts/Determinant/eGYqJ2TF">determinant</a> and <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a> of a nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a>.</li> <li>The only nilpotent <a href="/facts/Diagonalizable_matrix/y3MkyTCy">diagonalizable matrix</a> is the zero matrix.</li></ul> <p>See also: <a href="/facts/Jordan%25E2%2580%2593Chevalley_decomposition/w4PN88Xo">Jordan–Chevalley decomposition#Nilpotency criterion</a>. </p> <h2 id="classification">Classification</h2> <p>Consider the n × n {\displaystyle n\times n} (upper) <a href="/facts/Shift_matrix/P9QAUxoA">shift matrix</a>: </p> S = [ 0 1 0 … 0 0 0 1 … 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 … 1 0 0 0 … 0 ] . {\displaystyle S={\begin{bmatrix}0&1&0&\ldots &0\\0&0&1&\ldots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\ldots &1\\0&0&0&\ldots &0\end{bmatrix}}.} <p>This matrix has 1s along the <a href="/facts/Superdiagonal/39z9sIRb">superdiagonal</a> and 0s everywhere else. As a linear transformation, the shift matrix "shifts" the components of a vector one position to the left, with a zero appearing in the last position: </p> S ( x 1 , x 2 , … , x n ) = ( x 2 , … , x n , 0 ) . {\displaystyle S(x_{1},x_{2},\ldots ,x_{n})=(x_{2},\ldots ,x_{n},0).} <a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> <p>This matrix is nilpotent with degree n {\displaystyle n} , and is the <a href="/facts/Canonical_form/K2jhB44k">canonical</a> nilpotent matrix. </p><p>Specifically, if N {\displaystyle N} is any nilpotent matrix, then N {\displaystyle N} is <a href="/facts/Matrix_similarity/ySevpav1">similar</a> to a <a href="/facts/Block_diagonal_matrix/fPI8HX9f">block diagonal matrix</a> of the form </p> [ S 1 0 … 0 0 S 2 … 0 ⋮ ⋮ ⋱ ⋮ 0 0 … S r ] {\displaystyle {\begin{bmatrix}S_{1}&0&\ldots &0\\0&S_{2}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &S_{r}\end{bmatrix}}} <p>where each of the blocks S 1 , S 2 , … , S r {\displaystyle S_{1},S_{2},\ldots ,S_{r}} is a shift matrix (possibly of different sizes). This form is a special case of the <a href="/facts/Jordan_canonical_form/M7ezYbnl">Jordan canonical form</a> for matrices.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix </p> [ 0 1 0 0 ] . {\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}.} <p>That is, if N {\displaystyle N} is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b1, b2 such that <i>N</i>b1 = 0 and <i>N</i>b2 = b1. </p><p>This <a href="/facts/Classification_theorem/NWGgtNm6">classification theorem</a> holds for matrices over any <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>. (It is not necessary for the field to be algebraically closed.) </p> <h2 id="flag-of-subspaces">Flag of subspaces</h2> <p>A nilpotent transformation L {\displaystyle L} on R n {\displaystyle \mathbb {R} ^{n}} naturally determines a <a href="/facts/Flag_(linear_algebra)/gvDdt2TJ">flag</a> of subspaces </p> { 0 } ⊂ ker L ⊂ ker L 2 ⊂ … ⊂ ker L q − 1 ⊂ ker L q = R n {\displaystyle \{0\}\subset \ker L\subset \ker L^{2}\subset \ldots \subset \ker L^{q-1}\subset \ker L^{q}=\mathbb {R} ^{n}} <p>and a signature </p> 0 = n 0 < n 1 < n 2 < … < n q − 1 < n q = n , n i = dim ker L i . {\displaystyle 0=n_{0}<n_{1}<n_{2}<\ldots <n_{q-1}<n_{q}=n,\qquad n_{i}=\dim \ker L^{i}.} <p>The signature characterizes L {\displaystyle L} <a href="/facts/Up_to/ydz4ztzX">up to</a> an invertible <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a>. Furthermore, it satisfies the inequalities </p> n j + 1 − n j ≤ n j − n j − 1 , for all j = 1 , … , q − 1. {\displaystyle n_{j+1}-n_{j}\leq n_{j}-n_{j-1},\qquad {\mbox{for all }}j=1,\ldots ,q-1.} <p>Conversely, any sequence of natural numbers satisfying these inequalities is the signature of a nilpotent transformation. </p> <h2 id="additional-properties">Additional properties</h2> <ul><li>If N {\displaystyle N} is nilpotent of index k {\displaystyle k} , then I + N {\displaystyle I+N} and I − N {\displaystyle I-N} are <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a>, where I {\displaystyle I} is the n × n {\displaystyle n\times n} <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>. The inverses are given by ( I + N ) − 1 = ∑ m = 0 k ( − N ) m = I − N + N 2 − N 3 + N 4 − N 5 + N 6 − N 7 + ⋯ + ( − N ) k ( I − N ) − 1 = ∑ m = 0 k N m = I + N + N 2 + N 3 + N 4 + N 5 + N 6 + N 7 + ⋯ + N k {\displaystyle {\begin{aligned}(I+N)^{-1}&=\displaystyle \sum _{m=0}^{k}\left(-N\right)^{m}=I-N+N^{2}-N^{3}+N^{4}-N^{5}+N^{6}-N^{7}+\cdots +(-N)^{k}\\(I-N)^{-1}&=\displaystyle \sum _{m=0}^{k}N^{m}=I+N+N^{2}+N^{3}+N^{4}+N^{5}+N^{6}+N^{7}+\cdots +N^{k}\\\end{aligned}}} </li><li>If N {\displaystyle N} is nilpotent, then det ( I + N ) = 1. {\displaystyle \det(I+N)=1.} <p>Conversely, if A {\displaystyle A} is a matrix and </p> det ( I + t A ) = 1 {\displaystyle \det(I+tA)=1\!\,} for all values of t {\displaystyle t} , then A {\displaystyle A} is nilpotent. In fact, since p ( t ) = det ( I + t A ) − 1 {\displaystyle p(t)=\det(I+tA)-1} is a polynomial of degree n {\displaystyle n} , it suffices to have this hold for n + 1 {\displaystyle n+1} distinct values of t {\displaystyle t} .</li><li>Every <a href="/facts/Singular_matrix/fPqXk3V8">singular matrix</a> can be written as a product of nilpotent matrices.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li><li>A nilpotent matrix is a special case of a <a href="/facts/Convergent_matrix/xD7wzR9e">convergent matrix</a>.</li></ul> <h2 id="generalizations">Generalizations</h2> <p>A <a href="/facts/Linear_operator/Q1PBKNVV">linear operator</a> T {\displaystyle T} is locally nilpotent if for every vector v {\displaystyle v} , there exists a k ∈ N {\displaystyle k\in \mathbb {N} } such that </p> T k ( v ) = 0. {\displaystyle T^{k}(v)=0.\!\,} <p>For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence. </p> <h2 id="notes">Notes</h2> <ul><li>Beauregard, Raymond A.; Fraleigh, John B. (1973), <a href="https://archive.org/details/firstcourseinlin0000beau"><i>A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields</i></a>, Boston: <a href="/facts/Houghton_Mifflin_Co./byG66zxw">Houghton Mifflin Co.</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-395-14017-X</li> <li><a href="/facts/Israel_Nathan_Herstein/fWts4GeY">Herstein, I. N.</a> (1975), <i>Topics In Algebra</i> (2nd ed.), <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">John Wiley & Sons</a></li> <li>Nering, Evar D. (1970), <i>Linear Algebra and Matrix Theory</i> (2nd ed.), New York: <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">Wiley</a>, <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/76091646">76091646</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://planetmath.org/nilpotentmatrix">Nilpotent matrix</a> and <a href="https://planetmath.org/nilpotenttransformation">nilpotent transformation</a> on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Herstein (1975, p. 294) - Herstein, I. N. (1975), Topics In Algebra (2nd ed.), John Wiley & Sons <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Beauregard & Fraleigh (1973, p. 312) - Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Co., ISBN 0-395-14017-X <a href="https://archive.org/details/firstcourseinlin0000beau" target="_blank">https://archive.org/details/firstcourseinlin0000beau</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Herstein (1975, p. 268) - Herstein, I. N. (1975), Topics In Algebra (2nd ed.), John Wiley & Sons <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Nering (1970, p. 274) - Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646 <a href="https://lccn.loc.gov/76091646" target="_blank">https://lccn.loc.gov/76091646</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Mercer, Idris D. (31 October 2005). "Finding "nonobvious" nilpotent matrices" (PDF). idmercer.com. self-published; personal credentials: PhD Mathematics, Simon Fraser University. Retrieved 5 April 2023. <a href="http://www.idmercer.com/nilpotent.pdf" target="_blank">http://www.idmercer.com/nilpotent.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Beauregard & Fraleigh (1973, p. 312) - Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Co., ISBN 0-395-14017-X <a href="https://archive.org/details/firstcourseinlin0000beau" target="_blank">https://archive.org/details/firstcourseinlin0000beau</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Beauregard & Fraleigh (1973, pp. 312, 313) - Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Co., ISBN 0-395-14017-X <a href="https://archive.org/details/firstcourseinlin0000beau" target="_blank">https://archive.org/details/firstcourseinlin0000beau</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>R. Sullivan, Products of nilpotent matrices, Linear and Multilinear Algebra, Vol. 56, No. 3 <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> </ol>