Matrix Similarity#
Definition: Matrix Similarity
Two square matrices \(A, B \in F^{n \times n}\) are similar if there is an invertible matrix \(P \in F^{n \times n}\) such that
\[B = P^{-1} A P.\]
Theorem: Similarity \(\implies\) Equal Determinants
If two square matrices are similar, then they have the same determinant.
Proof
TODO
Theorem: Similarity \(\implies\) Equal Trace
If two square matrices are similar, then they have the same trace.
Proof
TODO
Theorem: Similarity \(\implies\) Eigenvalues
If two square matrices are similar, then they have the same eigenvalues. Furthermore, the algebraic multiplicity and geometric multiplicity of each eigenvalue are the same for both square matrices.
Proof
TODO