What is nilpotent matrix with example?

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).

Is every matrix nilpotent?

We can say that Nilpotent matrices are a subset of singular matrices. That is, All nilpotent matrices are singular. But, NOT all singular matrices are nilpotent.

What do u mean by nilpotent matrix?

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 .

How do you show something is nilpotent?

If Every Trace of a Power of a Matrix is Zero, then the Matrix is Nilpotent Let A be an n×n matrix such that tr(An)=0 for all n∈N.

Is the zero matrix A nilpotent matrix?

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.

Are triangular matrices nilpotent?

If all of the entries on the main diagonal of a (upper or lower) triangular matrix are also 0, the matrix is called strictly (upper or lower) triangular. All finite strictly triangular matrices are nilpotent of index n as a consequence of the Cayley-Hamilton theorem.

Which of the following is nilpotent matrix?

A square matrix A is said to be a nilpotent matrix of degree r, if r is the least positive integer such that Ar=0.

How do you check a matrix is nilpotent?

A matrix A is said to be nilpotent if A^n = 0(matrix) for some integer greater/equal to 1, and smallest such positive power n is termed as its index of nilpotence e.g. the 2×2 matrix with the rows as [0,1], and [0,0] is a nilpotent matrix of index 2. A matrix A is called idempotent if A^2 = A.

Are upper triangular matrices nilpotent?

Is symmetric matrix nilpotent?

There is a natural way of recognizing nilpotent matrices among the persymmetric ones – the strictly upper triangular persymmetric matrices are nilpotent. In n×n complex matrices there exists a symmetric nilpotent of rank n−1. , then 多 is simultaneously similar to the linear space of strictly upper triangular matrices.

Is null matrix A nilpotent matrix?

Nilpotent matrix is a square matrix, which on multiplying with itself results in a null matrix. The following are some of the important properties of nilpotent matrices. The nilpotent matrix is a square matrix of order n × n. The index of a nilpotent matrix having an order of n ×n is either n or a value lesser than n.

Is nilpotent matrix zero?

where A∗ denotes the conjugate transpose of A, that is A∗=ˉAT. A matrix A is said to be nilpotent if there exists a positive integer k such that Ak is the zero matrix. (a) Prove that if A is both normal and nilpotent, then A is the zero matrix. You may use the fact that every normal matrix is diagonalizable.

Is null matrix nilpotent?

Nilpotent Matrix is a square matrix such that the product of the matrix with itself is equal to a null matrix. A square matrix M of order n × n is termed as a nilpotent matrix if Mk = 0.

Is the identity matrix nilpotent?

Example The identity matrix is idempotent, because I2 = I · I = I. Definition 2. An n× n matrix B is called nilpotent if there exists a power of the matrix B which is equal to the zero matrix. This means that there is an index k such that Bk = O.