Properties of a Nilpotent Matrix

Following are some important properties of a nilpotent matrix:

• A nilpotent matrix is always a square matrix of order “n × n.”

• Nilpotency index of a nilpotent matrix of order “n × n” is always equal to either n or less than n.

• Both the trace and the determinant of a nilpotent matrix are always equal to zero.

• As the determinant of a nilpotent matrix is zero, it is not invertible.

• Null matrix is the only diagonalizable nilpotent matrix.

• A nilpotent matrix is a scalar matrix.

• Any triangular matrices with zeros on the principal diagonal are also nilpotent.

• Eigenvalues of a nilpotent matrix are always equal to zero.

Articles Related to Nilpotent Matrix:

• Determinant of a Matrix
• Inverse of a Matrix
• Matrix Addition and Scalar Multiplication

Nilpotent Matrices are special types of square matrices, they are special because the product of a Nilpotent Matrix with itself is equal to a null matrix. Let’s take a square matrix A of order n × n it is considered a nilpotent matrix if A^k = 0. Here k is always less than equal to n. 

In this article, we will learn about the Nilpotent Matrix in detail.