12.4.1. Similarity Transformations

Similarity transformations are particularly important to the eigenvalue problem because they preserve the eigenvalues as the matrix is changed to the desired shape.

Similar Matrices Theorem

Theorem 12.1 (Similar Matrices)

Matrices \(\bf{A}\) and \(\bf{B}\) are similar if there exists a matrix \(\bf{M}\) for which the relationship \(\mathbf{B} = \mathbf{M}^{-1}\,\mathbf{A\,M}\) holds. Since matrices \(\bf{A}\) and \(\bf{B}\) are similar, they have the same eigenvalues.

Proof. For \(\bm{x} \neq \bm{0}\), \(\m{A}\, \bm{x} = \lambda\, \bm{x}\), and non-singular matrix \(\m{M}\), there exists matrix \(\m{B} = \m{M}^{-1}\,\m{A\,M}\) such that \(\m{B}\,(\m{M}^{-1}\,\bm{x}) = \lambda\,(\m{M}^{-1}\,\bm{x})\).

\[\mathbf{A}\,(\mathbf{M\,M}^{-1})\,\bm{x} = \lambda\,\bm{x}\]

Now pre-multiply both sides by \(\mathbf{M}^{-1}\) and substitute for \(\bf{B}\).

\[\begin{split}\begin{aligned} &(\mathbf{M}^{-1}\mathbf{A\,M})\,\mathbf{M}^{-1}\,\bm{x} = \lambda\,\mathbf{M}^{-1}\,\bm{x} \\ &\mathbf{B}\,(\mathbf{M}^{-1}\,\bm{x}) = \lambda\,(\mathbf{M}^{-1}\,\bm{x}) \end{aligned}\end{split}\]

Thus, an eigenvalue of \(\bf{B}\) is \(\lambda\), and the corresponding eigenvector of \(\bf{B}\) is \(\mathbf{M}^{-1}\,\bm{x}\).

\(\qed\)

Algorithms that compute the eigenvalues of a matrix must maintain similarity. By the triangularization theorem (Schur Decomposition), a sequence of similarity transforms with unitary matrices, \(\m{Q}_i\), may be used to transform \(\m{A}\) into a triangular matrix \(\m{T} = \m{Q\, A\, Q}^H\) where \(\m{Q}\) is the product of the unitary matrices, \(\m{Q} = \m{Q}_0 \times \m{Q}_1 \times \cdots \times \m{Q}_k\). The eigenvalues of \(\m{A}\) are on the diagonal of \(\m{T}\), and \(\m{Q}\), the matrix of Schur vectors, may be used to find the eigenvectors of \(\m{A}\).