What is index of a nilpotent matrix?

A square matrix X is said to be nilpotent if Xr = 0 for some positive integer r. The least such positive integer is called the index (or, degree) of nilpotency.

Is nilpotent of index?

In linear algebra, a nilpotent matrix is a square matrix N such that \[{N^k} = 0\]. For some positive integer k. The smallest integer such k is called the index of N, sometimes the degree of N. A square matrix is a matrix with the same number of rows and columns.

How do you find the degree of nilpotent?

If A is an n×n matrix, then it is said to be nilpotent if A^m = O(zero matrix) for some positive integer m. Let k be the least such integer such that A^k=O. We then say that k is the degree of nilpotence of A. Hence A satisfies the polynomial equation x^k=0.

What is index in matrix?

An index matrix is a matrix with exactly one non-zero entry per row. Index matrices are useful for mapping observations to unique covariate values, for example.

How do you find if a matrix is nilpotent?

An n×n matrix A is called nilpotent if Ak=O, where O is the n×n zero matrix. Prove the followings. (a) The matrix A is nilpotent if and only if all the eigenvalues of A is zero. (b) The matrix A is nilpotent if and only if An=O.

How do you show a group is nilpotent?

24.3 Definition. A group G is nilpotent if Zi(G) = G for some i. If G is a nilpotent group then the nilpotency class of G is the smallest n 0 such that Zn(G) = G.

How do you check if a matrix is nilpotent or not?

Is nilpotent zero a matrix?

So no power of A can be the zero matrix. (b) By (a), a nilpotent matrix can have no nonzero eigenvalues, i.e., all its eigenvalues are 0.

Is nilpotent matrix zero?

A square matrix A is called nilpotent if some power of A is the zero matrix. Namely, A is nilpotent if there exists a positive integer k such that Ak=O, where O is the zero matrix.

How do you find the index and signature of a matrix?

Index: The index of the quadratic form is equal to the number of positive Eigen values of the matrix of quadratic form. Signature: The index of the quadratic form is equal to the difference between the number of positive Eigen values and the number of negative Eigen values of the matrix of quadratic form.