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:<math>\mathbf \tilde{x}_{\tilde{\lambda}=1} = \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix}, \quad \mathbf \tilde{x}_{\tilde{\lambda}=3} = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}.</math>
We observe that each one of the Ritz vectors is exactly one of the eigenvectors of <math>A</math> for the given <math>V</math> as well as the Ritz values give exactly two of the three eigenvalues of <math>A</math>. A mathematical explanation for the exact approximation is based on the fact that the [[column space]] of the matrix <math>V</math> happens to be exactly the same as the subspace spanned by the two eigenvectors <math>\mathbf x_{\lambda=1}</math> and <math>\mathbf x_{\lambda=3}</math> in this example.
== For matrix singular value problems ==
[[Singular_value_decomposition#Truncated_SVD | Truncated singular value decomposition (SVD)]] in numerical linear algebra can also use the '''Rayleigh–Ritz method''' to find approximations to left and right singular vectors of the matrix <math> M \in \mathbb{C}^{M \times N}</math> of size <math>M</math>-by-<math>N</math> in given subspaces by turning the singular value problem into an eigenvalue problem.
=== Using the normal matrix ===
The [[Hermitian matrix|Hermitian]] '''normal matrix''' <math> A = M^* M \in \mathbb{C}^{N \times N}</math> of size <math>N</math>-by-<math>N</math> allows directly applying the Rayleigh–Ritz method taking advantage of the fact that for a given <math>N</math>-by-<math>m</math> matrix <math> W \in \mathbb{C}^{N \times m} </math> with [[orthonormal]] columns the eigenvalue problem for the <math>m</math>-by-<math>m</math> matrix
:<math> W^* A W = W^* M^* M W = (M W)^* M W</math>
can be interpreted as a singular value problem for the <math>N</math>-by-<math>m</math> matrix <math>M W</math>. This interpretation allows simple calculation of both left and right approximate singular vectors as follows.
# Compute the <math> N \times m </math> matrix <math> M W </math>
# Compute the [[Singular_value_decomposition#Thin_SVD|thin, or economy-sized, SVD]] <math> M W = \mathbf {U}{{\Sigma }}\mathbf {V}^{*},</math> with <math>N</math>-by-<math>m</math> matrix <math>\mathbf {U}</math>, <math>m</math>-by-<math>m</math> diagonal matrix <math>{\Sigma}</math>, and <math>m</math>-by-<math>m</math> matrix <math>\mathbf {V}</math>.
# Compute the matrices of the Ritz left <math>U = \mathbf {U}</math> and right <math>V = \mathbf {V} W^*</math> singular vectors
# Output approximations <math>U, \Sigma, V</math>, called the Ritz singular triplets, to selected singular values and the corresponding left and right singular vectors of the original matrix <math>M</math> representing an approximate [[Singular_value_decomposition#Truncated_SVD | Truncated singular value decomposition (SVD)]]
== Derivation from calculus of variations ==
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