QR Decomposition Structure
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A QR Decomposition Structure is a matrix decomposition structure composed of an orthogonal matrix [math]\displaystyle{ Q }[/math] and an upper triangular matrix [math]\displaystyle{ R }[/math].
- AKA: QR Factorization.
- Context:
- It can be produced by a QR Decomposition System (that solves a QR decomposition task).
- It can (often) be used to solve a Linear Least Squares Optimization Task.
- It can be computed by a Gram-Schmidt Process and Householder Transformations.
- Its computation is numerically stable because it proceeds by Orthogonal Transformations.
- Example(s):
- [math]\displaystyle{ \begin{bmatrix}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & -\frac{1}{\sqrt{5}} \end{bmatrix} \begin{bmatrix}\sqrt{5} & \frac{11}{\sqrt{5}} \\ 0 & \frac{2}{\sqrt{5}} \end{bmatrix} }[/math], for [math]\displaystyle{ \operatorname{QR} \begin{pmatrix}\begin{bmatrix}1 & 3 \\ 2 & 4 \end{bmatrix}\end{pmatrix} }[/math].
- [math]\displaystyle{ \begin{bmatrix}\frac{1}{\sqrt{35}} & \frac{13}{\sqrt{210}} \\ \frac{3}{\sqrt{35}} & 2\sqrt{\frac{2}{105}} \\ \sqrt{\frac{5}{7}} & -\sqrt{\frac{5}{42}} \end{bmatrix} \begin{bmatrix}\sqrt{35} & \frac{44}{\sqrt{35}} \\ 0 & 2\sqrt{\frac{6}{35}} \end{bmatrix} }[/math], for [math]\displaystyle{ \operatorname{QR}\begin{pmatrix}\begin{bmatrix}1 & 2 \\ 3 & 4 \\ 5 & 6\end{bmatrix}\end{pmatrix} }[/math].
- …
- [math]\displaystyle{ \begin{bmatrix}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & -\frac{1}{\sqrt{5}} \end{bmatrix} \begin{bmatrix}\sqrt{5} & \frac{11}{\sqrt{5}} \\ 0 & \frac{2}{\sqrt{5}} \end{bmatrix} }[/math], for [math]\displaystyle{ \operatorname{QR} \begin{pmatrix}\begin{bmatrix}1 & 3 \\ 2 & 4 \end{bmatrix}\end{pmatrix} }[/math].
- Counter-Example(s):
- See: Linear Span, Linear Least Squares, Eigenvalue Algorithm, QR Algorithm, Linearly Independent, Orthonormal Basis, Column Space.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/QR_decomposition Retrieved:2015-3-3.
- In linear algebra, a QR decomposition (also called a QR factorization) of a matrix is a decomposition of a matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the QR algorithm.
If A has n linearly independent columns, then the first n columns of Q form an orthonormal basis for the column space of A. More specifically, the first k columns of Q form an orthonormal basis for the span of the first k columns of A for any 1 ≤ k ≤ n.[1] The fact that any column k of A only depends on the first k columns of Q is responsible for the triangular form of R.
- In linear algebra, a QR decomposition (also called a QR factorization) of a matrix is a decomposition of a matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the QR algorithm.
- ↑ L. N. Trefethen and D. Bau, Numerical Linear Algebra (SIAM, 1997).
2017
- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/QR_decomposition Retrieved:2017-11-15.
- In linear algebra, a QR decomposition (also called a QR factorization) of a matrix is a decomposition of a matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the QR algorithm.