Matrix Similarity - Properties

Properties

Similarity is an equivalence relation on the space of square matrices.

Similar matrices share many properties:

• Rank
• Determinant
• Trace
• Eigenvalues (though the eigenvectors will in general be different)
• Characteristic polynomial
• Minimal polynomial (among the other similarity invariants in the Smith normal form)
• Elementary divisors

There are two reasons for these facts:

• Two similar matrices can be thought of as describing the same linear map, but with respect to different bases
• The map X ↦ P−1XP is an automorphism of the associative algebra of all n-by-n matrices, as the one-object case of the above category of all matrices.

Because of this, for a given matrix A, one is interested in finding a simple "normal form" B which is similar to A—the study of A then reduces to the study of the simpler matrix B. For example, A is called diagonalizable if it is similar to a diagonal matrix. Not all matrices are diagonalizable, but at least over the complex numbers (or any algebraically closed field), every matrix is similar to a matrix in Jordan form. Another normal form, the rational canonical form, works over any field. By looking at the Jordan forms or rational canonical forms of A and B, one can immediately decide whether A and B are similar. The Smith normal form can be used to determine whether matrices are similar, though unlike the Jordan and rational canonical forms, a matrix is not necessarily similar to its Smith normal form.