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If the subspace with the orthonormal basis given by the columns of the matrix <math> V \in \mathbb{C}^{N \times m} </math> contains <math> k \leq m </math> vectors that are close to eigenvectors of the matrix <math>A</math>, the '''Rayleigh–Ritz method''' above finds <math>k</math> Ritz vectors that well approximate these eigenvectors. The easily computable quantity <math> \| A \tilde{\mathbf{x}}_i - \tilde{\lambda}_i \tilde{\mathbf{x}}_i\|</math> determines the accuracy of such an approximation for every Ritz pair.
In the easiest case <math>m = 1</math>, the <math> N \times m </math> matrix <math>V</math> turns into a unit column-vector <math>v</math>, the <math> m \times m </math> matrix <math> V^* A V </math> is a scalar that is equal to the [[Rayleigh quotient]] <math>\rho(v) = v^*Av/v^*v</math>, the only <math>i = 1</math> solution to the eigenvalue problem is <math>y_i = 1</math> and <math>\mu_i = \rho(v)</math>, and the only
Another useful connection to the [[Rayleigh quotient]] is that <math>\mu_i = \rho(v_i)</math> for every Ritz pair <math>(\tilde{\lambda}_i, \tilde{\mathbf{x}}_i)</math>, allowing to derive some properties of Ritz values <math>\mu_i</math> from the corresponding theory for the [[Rayleigh quotient]]. For example, if <math>A</math> is a [[Hermitian matrix]], its [[Rayleigh quotient]] (and thus its every Ritz value) is real and takes values within the closed interval of the smallest and largest eigenvalues of <math>A</math>.
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