Properties of Similar Matrices

Following are some important properties of similar matrices A and B:

• Ranks of two similar matrices are the same, i.e., the rank of A = rank of B.
• Determinants of two similar matrices are the same, i.e., det (A) = det (B).
• Trace of two similar matrices is the same, i.e., tr(A) = tr(B).
• Eigenvalues of two similar matrices are the same, but their eigenvectors are normally different.
• If A and B are two similar matrices, then Aⁿ and Bⁿ are also similar matrices.
• A matrix and its transpose matrix are similar, i.e., A ∼ A^T.
• Two similar matrices, A and B, are said to have the same characteristic polynomial.
• A matrix “A” is similar to itself, i.e., the similarity of matrices is reflexive.

A = I^-1AI,

where the identity matrix “I” is the change-of-basis matrix.

• If matrix A is similar to matrix B, then matrix B is said to be similar to matrix A, i.e., the similarity of matrices is symmetric.

P^-1AP = B

A = PBP^-1

• If matrix A is similar to matrix B and matrix B is similar to matrix C, then matrix A is said to be similar to matrix C, i.e., the similarity of matrices is transitive.

Also Check

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

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.