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Nilpotent matrix
Endomorphism whose sufficiently large power is zero

In linear algebra, a nilpotent matrix is a square matrix N such that

N k = 0 {\displaystyle N^{k}=0\,}

for some positive integer k {\displaystyle k} . The smallest such k {\displaystyle k} is called the index of N {\displaystyle N} , sometimes the degree of N {\displaystyle N} .

More generally, a nilpotent transformation is a linear transformation L {\displaystyle L} of a vector space such that L k = 0 {\displaystyle L^{k}=0} for some positive integer k {\displaystyle k} (and thus, L j = 0 {\displaystyle L^{j}=0} for all j ≥ k {\displaystyle j\geq k} ). Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.

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Examples

Example 1

The matrix

A = [ 0 1 0 0 ] {\displaystyle A={\begin{bmatrix}0&1\\0&0\end{bmatrix}}}

is nilpotent with index 2, since A 2 = 0 {\displaystyle A^{2}=0} .

Example 2

More generally, any n {\displaystyle n} -dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index ≤ n {\displaystyle \leq n} . For example, the matrix

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}}}

is nilpotent, with

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}}}

The index of B {\displaystyle B} is therefore 4.

Example 3

Although the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example,

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}}}

although the matrix has no zero entries.

Example 4

Additionally, any matrices of the form

[ 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}}}

such as

[ 5 5 5 6 6 6 − 11 − 11 − 11 ] {\displaystyle {\begin{bmatrix}5&5&5\\6&6&6\\-11&-11&-11\end{bmatrix}}}

or

[ 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}}}

square to zero.

Example 5

Perhaps some of the most striking examples of nilpotent matrices are n × n {\displaystyle n\times n} square matrices of the form:

[ 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}}}

The first few of which are:

[ 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 }

These matrices are nilpotent but there are no zero entries in any powers of them less than the index.5

Example 6

Consider the linear space of polynomials of a bounded degree. The derivative 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.

Characterization

For an n × n {\displaystyle n\times n} square matrix N {\displaystyle N} with real (or complex) entries, the following are equivalent:

  • N {\displaystyle N} is nilpotent.
  • The characteristic polynomial for N {\displaystyle N} is det ( x I − N ) = x n {\displaystyle \det \left(xI-N\right)=x^{n}} .
  • The minimal polynomial for N {\displaystyle N} is x k {\displaystyle x^{k}} for some positive integer k ≤ n {\displaystyle k\leq n} .
  • The only complex eigenvalue for N {\displaystyle N} is 0.

The last theorem holds true for matrices over any field of characteristic 0 or sufficiently large characteristic. (cf. Newton's identities)

This theorem has several consequences, including:

  • 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.
  • The determinant and trace of a nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be invertible.
  • The only nilpotent diagonalizable matrix is the zero matrix.

See also: Jordan–Chevalley decomposition#Nilpotency criterion.

Classification

Consider the n × n {\displaystyle n\times n} (upper) shift matrix:

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}}.}

This matrix has 1s along the superdiagonal 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:

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).} 6

This matrix is nilpotent with degree n {\displaystyle n} , and is the canonical nilpotent matrix.

Specifically, if N {\displaystyle N} is any nilpotent matrix, then N {\displaystyle N} is similar to a block diagonal matrix of the form

[ 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}}}

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 Jordan canonical form for matrices.7

For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix

[ 0 1 0 0 ] . {\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}.}

That is, if N {\displaystyle N} is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b1, b2 such that Nb1 = 0 and Nb2 = b1.

This classification theorem holds for matrices over any field. (It is not necessary for the field to be algebraically closed.)

Flag of subspaces

A nilpotent transformation L {\displaystyle L} on R n {\displaystyle \mathbb {R} ^{n}} naturally determines a flag of subspaces

{ 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}}

and a signature

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}.}

The signature characterizes L {\displaystyle L} up to an invertible linear transformation. Furthermore, it satisfies the inequalities

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.}

Conversely, any sequence of natural numbers satisfying these inequalities is the signature of a nilpotent transformation.

Additional properties

  • If N {\displaystyle N} is nilpotent of index k {\displaystyle k} , then I + N {\displaystyle I+N} and I − N {\displaystyle I-N} are invertible, where I {\displaystyle I} is the n × n {\displaystyle n\times n} identity matrix. 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}}}
  • If N {\displaystyle N} is nilpotent, then det ( I + N ) = 1. {\displaystyle \det(I+N)=1.}

    Conversely, if A {\displaystyle A} is a matrix and

    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} .
  • Every singular matrix can be written as a product of nilpotent matrices.8
  • A nilpotent matrix is a special case of a convergent matrix.

Generalizations

A linear operator 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

T k ( v ) = 0. {\displaystyle T^{k}(v)=0.\!\,}

For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.

Notes

References

  1. Herstein (1975, p. 294) - Herstein, I. N. (1975), Topics In Algebra (2nd ed.), John Wiley & Sons ↩

  2. 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 https://archive.org/details/firstcourseinlin0000beau ↩

  3. Herstein (1975, p. 268) - Herstein, I. N. (1975), Topics In Algebra (2nd ed.), John Wiley & Sons ↩

  4. Nering (1970, p. 274) - Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646 https://lccn.loc.gov/76091646 ↩

  5. 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. http://www.idmercer.com/nilpotent.pdf ↩

  6. 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 https://archive.org/details/firstcourseinlin0000beau ↩

  7. 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 https://archive.org/details/firstcourseinlin0000beau ↩

  8. R. Sullivan, Products of nilpotent matrices, Linear and Multilinear Algebra, Vol. 56, No. 3 ↩