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Remark It is left as an exercise to show that if a matrix B is similar to a matrix A, then necessarily A is similar to B. Therefore,
we shall usually simply say that A and B are similar.
Similarity Invariants
Similar matrices often have properties in common; for example, if A and B are similar matrices, then A and B have the same
determinant. To see that this is so, suppose that
Then
We make the following definition.
DEFINITION
A property of square matrices is said to be a similarity invariant or invariant under similarity if that property is shared by
any two similar matrices.
In the terminology of this definition, the determinant of a square matrix is a similarity invariant. Table 1 lists some other
important similarity invariants. The proofs of some of the results in Table 1 are given in the exercises.
Table 1
Similarity Invariants
Property Description
Determinant A and have the same determinant.
Invertibility A is invertible if and only if is invertible.
Rank A and have the same rank.
Nullity A and have the same nullity.
Trace A and have the same trace.
Characteristic A and have the same characteristic polynomial.
polynomial
Eigenvalues A and have the same eigenvalues.
Eigenspace If λ is an eigenvalue of A and , then the eigenspace of A corresponding to λ and the
dimension
eigenspace of corresponding to λ have the same dimension.
It follows from Theorem 8.5.2 that two matrices representing the same linear operator with respect to different
