Nilpotent matrix

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

for some positive integer k. The smallest such k is sometimes called the degree of N.

More generally, a nilpotent transformation is a linear transformation L of a vector space such that L^k = 0 for some positive integer k. Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.

Contents 1 Examples 2 Characterization 3 Classification 4 Flag of subspaces 5 Additional properties 6 Generalizations 7 References 8 External links

[edit] Examples

The matrix

is nilpotent, since M² = 0. More generally, any triangular matrix with 0's along the main diagonal is nilpotent. For example, the matrix

is nilpotent, with

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

both square to zero, though neither matrix has zero entries.

[edit] Characterization

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

1. N is nilpotent.
2. The minimal polynomial for N is λ^k for some positive integer k.
3. The characteristic polynomial for N is λⁿ.
4. The only (complex) eigenvalue for N is 0.

The same theorem holds true for matrices over any algebraically closed field.

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.
• A 2 × 2 matrix is nilpotent if and only if both the determinant and trace are zero.
• The only nilpotent diagonalizable matrix is the zero matrix.

[edit] Classification

Consider the n × n shift matrix:

This matrix has 1's along the superdiagonal and 0's everywhere else. As a linear transformation, the shift matrix “shifts” the components of a vector one slot to the left:

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

where each of the blocks S₁, S₂, ..., S_r is a shift matrix. This theorem is a special case of the Jordan canonical form for matrices.

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

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.)

[edit] Flag of subspaces

A nilpotent transformation L on Rⁿ naturally determines a flag of subspaces

and a signature

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

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

[edit] Additional properties

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

where only finitely many terms of this sum are nonzero.

• If N is nilpotent, then

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

for all values of t, then A is nilpotent.

• Every singular matrix can be written as a product of nilpotent matrices.^[1]

[edit] Generalizations

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

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

[edit] References

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

[edit] External links

• Nilpotent matrix and nilpotent transformation on PlanetMath.