How do you prove a matrix is nilpotent?

How do you prove a matrix is nilpotent?

Problem 77. A square matrix A is called nilpotent if there exists a positive integer k such that Ak=O, where O is the zero matrix. (a) If A is a nilpotent n×n matrix and B is an n×n matrix such that AB=BA.

How do you find the order of a nilpotent matrix?

By definition, a nilpotent matrix A satisfies for some positive integer k, and the smallest such k is the index. So, just keep multiplying copies of A until you get 0. You’re even guaranteed that the index is at most n for an matrix.

What are the possible eigenvalues of a nilpotent matrix?

(b) By (a), a nilpotent matrix can have no nonzero eigenvalues, i.e., all its eigenvalues are 0.

What is index in nilpotent matrix?

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From Wikipedia, the free encyclopedia. In linear algebra, a nilpotent matrix is a square matrix N such that. for some positive integer . The smallest such is called the index of , sometimes the degree of .

Why are nilpotent matrix not diagonalizable?

Since A is nilpotent the minimalpolynom is tk for a k∈N. The minimalpolynom of a diagonalisable matrix is a product of distinct linear factors. So it must be t for a diagonalisable nilpotent matrix and so A=0, because every matrix is a zero of its minimalpolynom.

What is example of nilpotent matrix?

Examples of Nilpotent Matrix A n-dimensional triangular matrix with zeros along the main diagonal can be taken as a nilpotent matrix. 3. Also, a matrix without any zeros can also be referred as a nilpotent matrix. The following is a general form of a non-zero matrix, which is a nilpotent matrix.

What is the use of nilpotent matrix?

Examples & Properties. A nilpotent matrix (P) is a square matrix, if there exists a positive integer ‘m’ such that Pm = O. In other words, matrix P is called nilpotent of index m or class m if Pm = O and Pm-1 ≠ O. Here O is the null matrix (or zero matrix).

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What rank order means?

Noun. 1. rank order – an arrangement according to rank. ordering, order – the act of putting things in a sequential arrangement; “there were mistakes in the ordering of items on the list”