Nilpotent matrix

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

N^{k}=0\,

for some positive integer k. The smallest such k is sometimes called the degree or index of N.^[1]

More generally, a nilpotent transformation is a linear transformation L of a vector space such that L^k = 0 for some positive integer k (and thus, L^j = 0 for all j ≥ k).^[2]^[3]^[4] Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.

Examples

The matrix

M={\begin{bmatrix}0&1\\0&0\end{bmatrix}}

is nilpotent, since M² = 0. More generally, any triangular matrix with 0s along the main diagonal is nilpotent, with degree $\leq n$ . For example, the matrix

N={\begin{bmatrix}0&2&1&6\\0&0&1&2\\0&0&0&3\\0&0&0&0\end{bmatrix}}

is nilpotent, with

N^{2}={\begin{bmatrix}0&0&2&7\\0&0&0&3\\0&0&0&0\\0&0&0&0\end{bmatrix}};\ N^{3}={\begin{bmatrix}0&0&0&6\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}};\ N^{4}={\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}}.

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

N={\begin{bmatrix}5&-3&2\\15&-9&6\\10&-6&4\end{bmatrix}}

squares to zero, though the matrix has no zero entries.

Characterization

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

N is nilpotent.
The minimal polynomial for N is x^k for some positive integer k ≤ n.
The characteristic polynomial for N is xⁿ.
The only eigenvalue for N is 0.^[5]
tr(N^k) = 0 for all k > 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 degree of an n × n nilpotent matrix is always less than or equal to n. For example, every 2 × 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.

Classification

Consider the n × n shift matrix:

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},\ldots ,x_{n})=(x_{2},\ldots ,x_{n},0).

^[6]

This matrix is nilpotent with degree n, and is the “canonical” nilpotent matrix.

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

{\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 S₁, S₂, ..., S_r is a shift matrix (possibly of different sizes). This form is a special case of the Jordan canonical form for matrices.^[7]^[8]

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

{\begin{bmatrix}0&1\\0&0\end{bmatrix}}.

That is, if 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 on Rⁿ naturally determines a flag of subspaces

\{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}<\ldots <n_{q-1}<n_{q}=n,\qquad n_{i}=\dim \ker L^{i}.

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

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 is nilpotent, then I + N is invertible, where I is the n × n identity matrix. The inverse is given by

(I+N)^{-1}=I-N+N^{2}-N^{3}+\cdots ,

where only finitely many terms of this sum are nonzero.

If N is nilpotent, then

\det(I+N)=1,\!\,

where I denotes the n × n identity matrix. Conversely, if A is a matrix and

\det(I+tA)=1\!\,

for all values of t, then A is nilpotent. In fact, since

p(t)=\det(I+tA)-1

is a polynomial of degree

n

, it suffices to have this hold for

n+1

distinct values of

t

Every singular matrix can be written as a product of nilpotent matrices.^[9]
A nilpotent matrix is a special case of a convergent matrix.

Generalizations

A linear operator T is locally nilpotent if for every vector v, there exists a k such that

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

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

Notes

↑ Herstein (1964, p. 250)
↑ Beauregard & Fraleigh (1973, p. 312)
↑ Herstein (1964, p. 224)
↑ Nering (1970, p. 274)
↑ Herstein (1964, p. 248)
↑ Beauregard & Fraleigh (1973, p. 312)
↑ Beauregard & Fraleigh (1973, pp. 312,313)
↑ Herstein (1964, p. 250)
↑ R. Sullivan, Products of nilpotent matrices, Linear and Multilinear Algebra, Vol. 56, No. 3

References

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
Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016
Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646

External links

Nilpotent matrix and nilpotent transformation on PlanetMath.

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