Editing Orthogonal matrix (section)

===Decompositions===
A number of important [[matrix decomposition]]s {{harv|Golub|Van Loan|1996}} involve orthogonal matrices, including especially:

;[[QR decomposition|{{mvar|QR}} decomposition]] : {{math|1=''M'' = ''QR''}}, {{mvar|Q}} orthogonal, {{mvar|R}} upper triangular
;[[Singular value decomposition]] : {{math|1=''M'' = ''U''Σ''V''<sup>T</sup>}}, {{mvar|U}} and {{mvar|V}} orthogonal, {{math|Σ}} diagonal matrix
;[[Eigendecomposition of a matrix|Eigendecomposition of a symmetric matrix]] (decomposition according to the [[spectral theorem]]) : {{math|1=''S'' = ''Q''Λ''Q''<sup>T</sup>}}, {{mvar|S}} symmetric, {{mvar|Q}} orthogonal, {{math|Λ}} diagonal
;[[Polar decomposition]] : {{math|1=''M'' = ''QS''}}, {{mvar|Q}} orthogonal, {{mvar|S}} symmetric positive-semidefinite

====Examples====
Consider an [[overdetermined system of linear equations]], as might occur with repeated measurements of a physical phenomenon to compensate for experimental errors. Write {{math|1=''A'''''x''' = '''b'''}}, where {{mvar|A}} is {{math|''m'' × ''n''}}, {{math|''m'' > ''n''}}.
A {{mvar|QR}} decomposition reduces {{mvar|A}} to upper triangular {{mvar|R}}. For example, if {{mvar|A}} is {{nowrap|5 × 3}} then {{mvar|R}} has the form
<math display="block">R = \begin{bmatrix}
\cdot & \cdot & \cdot \\
0 & \cdot & \cdot \\
0 & 0 & \cdot \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}.</math>

The [[linear least squares (mathematics)|linear least squares]] problem is to find the {{math|'''x'''}} that minimizes {{math|{{norm|''A'''''x''' − '''b'''}}}}, which is equivalent to projecting {{math|'''b'''}} to the subspace spanned by the columns of {{mvar|A}}. Assuming the columns of {{mvar|A}} (and hence {{mvar|R}}) are independent, the projection solution is found from {{math|1=''A''<sup>T</sup>''A'''''x''' = ''A''<sup>T</sup>'''b'''}}. Now {{math|''A''<sup>T</sup>''A''}} is square ({{math|''n'' × ''n''}}) and invertible, and also equal to {{math|''R''<sup>T</sup>''R''}}. But the lower rows of zeros in {{mvar|R}} are superfluous in the product, which is thus already in lower-triangular upper-triangular factored form, as in [[Gaussian elimination]] ([[Cholesky decomposition]]). Here orthogonality is important not only for reducing {{math|1=''A''<sup>T</sup>''A'' = (''R''<sup>T</sup>''Q''<sup>T</sup>)''QR''}} to {{math|''R''<sup>T</sup>''R''}}, but also for allowing solution without magnifying numerical problems.

In the case of a linear system which is underdetermined, or an otherwise non-[[invertible matrix]], singular value decomposition (SVD) is equally useful. With {{mvar|A}} factored as {{math|''U''Σ''V''<sup>T</sup>}}, a satisfactory solution uses the Moore-Penrose [[pseudoinverse]], {{math|''V''Σ<sup>+</sup>''U''<sup>T</sup>}}, where {{math|Σ<sup>+</sup>}} merely replaces each non-zero diagonal entry with its reciprocal. Set {{math|'''x'''}} to {{math|''V''Σ<sup>+</sup>''U''<sup>T</sup>'''b'''}}.

The case of a square invertible matrix also holds interest. Suppose, for example, that {{mvar|A}} is a {{nowrap|3 × 3}} rotation matrix which has been computed as the composition of numerous twists and turns. Floating point does not match the mathematical ideal of real numbers, so {{mvar|A}} has gradually lost its true orthogonality. A [[Gram–Schmidt process]] could [[orthogonalization|orthogonalize]] the columns, but it is not the most reliable, nor the most efficient, nor the most invariant method. The [[polar decomposition]] factors a matrix into a pair, one of which is the unique ''closest'' orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular. (Closeness can be measured by any [[matrix norm]] invariant under an orthogonal change of basis, such as the spectral norm or the Frobenius norm.) For a near-orthogonal matrix, rapid convergence to the orthogonal factor can be achieved by a "[[Newton's method]]" approach due to {{harvtxt|Higham|1986}} ([[#CITEREFHigham1990|1990]]), repeatedly averaging the matrix with its inverse transpose. {{harvtxt|Dubrulle|1999}} has published an accelerated method with a convenient convergence test.

For example, consider a non-orthogonal matrix for which the simple averaging algorithm takes seven steps
<math display="block">\begin{bmatrix}3 & 1\\7 & 5\end{bmatrix}
\rightarrow
\begin{bmatrix}1.8125 & 0.0625\\3.4375 & 2.6875\end{bmatrix}
\rightarrow \cdots \rightarrow
\begin{bmatrix}0.8 & -0.6\\0.6 & 0.8\end{bmatrix}</math>
and which acceleration trims to two steps (with {{mvar|γ}} = 0.353553, 0.565685).

<math display="block">\begin{bmatrix}3 & 1\\7 & 5\end{bmatrix}
\rightarrow
\begin{bmatrix}1.41421 & -1.06066\\1.06066 & 1.41421\end{bmatrix}
\rightarrow
\begin{bmatrix}0.8 & -0.6\\0.6 & 0.8\end{bmatrix}</math>

Gram-Schmidt yields an inferior solution, shown by a Frobenius distance of 8.28659 instead of the minimum 8.12404.

<math display="block">\begin{bmatrix}3 & 1\\7 & 5\end{bmatrix}
\rightarrow
\begin{bmatrix}0.393919 & -0.919145\\0.919145 & 0.393919\end{bmatrix}</math>