In mathematics, a nilpotent matrix is an n×n square matrix M such that For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For the square matrix section, see square matrix. ...

for some positive integer q. Similarly, a nilpotent transformation is a linear transformation L with L^q = 0 for some integer q. In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

These are special cases of a more general concept of nilpotence that applies not only to matrices and linear transformations but to members of rings. In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...

Examples

Consider the following matrix:

This is an example of a 4×4 nilpotent matrix (in fact, matrices of this form are called shift matrices). Notice the non-zero superdiagonal. The characteristic feature of this matrix is: In mathematics, a shift matrix is a binary matrix with ones only on the superdiagonal or subdiagonal, and zeroes elsewhere. ... In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology. ...

The super-diagonal keeps 'shifting' diagonally up, until one gets the null matrix. In mathematics, a zero matrix is a matrix with all its entries being zero. ...

The corresponding nilpotent transformation L : R⁴ → R⁴ is defined by:

There is a classification theorem showing that this is typical: a nilpotent matrix is similar to a block matrix, with diagonal square blocks generalizing this type, and other blocks zero. Several equivalence relations in mathematics are called similarity. ... In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ...

Properties

Let M be an n×n nilpotent matrix.

• The smallest integer q such that M^q = 0 is smaller than or equal to n.
• Over an algebraically closed field, a matrix M is nilpotent if and only if its eigenvalues are all zero. Therefore the determinant and trace of M are both zero, and nilpotent matrices are not invertible.
• Suppose A and B are matrices. If A is invertible, then A ^{− 1}B is nilpotent if and only if det(A + tB) does not depend on t. This follows since

when are eigenvalues of A ^{− 1}B.

• The characteristic polynomial of M is λⁿ.
• Every strictly upper triangular matrix or strictly lower triangular matrix is nilpotent.
• Every singular matrix can be written as a product of nilpotent matrices.^[1]

In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In linear algebra, an n-by-n (square) matrix A is called invertible or non-singular if there exists an n-by-n matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. ... In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. ... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

Classification theorem

The above example is typical, as the following result shows. Every nilpotent matrix is similar to a block diagonal matrix Several equivalence relations in mathematics are called similarity. ... In mathematics, a block diagonal matrix is a block matrix which is a square matrix, and having main diagonal blocks square matrices, such that the off-diagonal blocks are zero matrices. ...

where the blocks N_i have ones on the superdiagonal and zeros everywhere else:

This fact follows from the Jordan decomposition theorem, together with the result that every matrix similar to a nilpotent matrix is also nilpotent. In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K containing the eigenvalues of M, to what extent can M...

Flag of subspaces

A nilpotent transformation L on Rⁿ naturally determines a flag of subspaces In mathematics, a flag is an increasing sequence of subspaces of a vector space. ...

and a signature

The signature characterizes L up to an invertible linear transformation. Furthermore, it satisfies the inequalities Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

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

References

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

External links

• Nilpotent matrix and nilpotent transformation on PlanetMath.