.. _similar:

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.

.. _th-similar:

.. rubric:: Similar Matrices Theorem

.. prf:theorem::  Similar Matrices
   :label: similarMatrices

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

.. prf:proof::

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

   .. math:: \mathbf{A}\,(\mathbf{M\,M}^{-1})\,\bm{x} = \lambda\,\bm{x}

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

   .. math::

      \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}

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

   :math:`\qed`

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

.. index:: similar matrices theorem