Similar Matrices

Two square matrices A and B of the same order are said to be similar, if and only if there exists an invertible matrix “P” of the same order as A and B such that:

P^-1AP = B

The transformation of the matrix A into “P^-1AP” is called similarity transformation or conjugation by “P,” as we are transforming the matrix “A” into the matrix “B.” Here, the matrix “P” is known as the change-of-basis matrix. If two matrices A and B are said to be similar, then they are expressed as A ∼ B.

Examples of Similar Matrices

The matrices given below are similar matrices of order “2 × 2” through the invertible matrix P of the same order.

 and

 are similar matrices through the invertible matrix P.

A “matrix” is referred to as a rectangular array of numbers that are arranged in rows and columns. The horizontal lines are said to be rows, while the vertical lines are said to be columns. We can determine the size of a matrix by the number of rows and columns in it. If a matrix has “m” rows and “n” columns, then it is said to be an “m by n” matrix and is written as an “m × n” matrix.

For example, a matrix with five rows and three columns is a “5 × 3” matrix. We have various types of matrices, like rectangular, square, triangular, symmetric, singular, etc. In this article, we learn about similar matrices, their examples, and their properties.