From Wakapon
Jump to: navigation, search
Line 23: Line 23:
  
 
Where:
 
Where:
* '''A''' is an m by n real or complex matrix
+
* '''A''' is an <math>m \times n</math> real or complex matrix
* '''U''' is an m by n real or complex unitary matrix
+
* '''U''' is an <math>m \times m</math> real or complex unitary matrix
* <math>\bold{\Sigma}</math> is a m by n rectangular diagonal matrix with non-negative real numbers on the diagonal
+
* <math>\bold{\Sigma}</math> is an <math>m \times n</math> rectangular diagonal matrix with non-negative real numbers on the diagonal
* '''V''' is an n by n real or complex unitary matrix.
+
* '''V''' is an <math>n \times n</math> real or complex unitary matrix.
  
  

Revision as of 17:21, 30 July 2017

Matrix Decomposition Methods

QR Decomposition

QR Decomposition, or QR Factorization is the process of decomposing a matrix A this way: <math>\left [ \bold{A} \right ] = \left [ \bold{Q} \right ] \cdot \left [ \bold{R} \right ]</math>

Where:

  • Q is an orthogonal matrix
  • R is an upper triangular matrix


Applications

QR decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the QR algorithm.


SVD Decomposition

Singular Value Decomposition (SVD) is the process of decomposing a matrix A this way:

<math>\left [ \bold{A} \right ] = \left [ \bold{U} \right ] \cdot \left [ \bold{\Sigma} \right ] \cdot \left [ \bold{V} \right ]^T</math>

SVD.png

Where:

  • A is an <math>m \times n</math> real or complex matrix
  • U is an <math>m \times m</math> real or complex unitary matrix
  • <math>\bold{\Sigma}</math> is an <math>m \times n</math> rectangular diagonal matrix with non-negative real numbers on the diagonal
  • V is an <math>n \times n</math> real or complex unitary matrix.


The diagonal entries <math>\sigma_i</math> of <math>\bold{\Sigma}</math> are known as the singular values of A. The columns of U and the columns of V are called the left-singular vectors and right-singular vectors of A, respectively.


The singular value decomposition can be computed using the following observations:

  • The left-singular vectors of M are a set of orthonormal eigenvectors of <math>\bold{A}\bold{A}^T</math>.
  • The right-singular vectors of M are a set of orthonormal eigenvectors of <math>\bold{A}^T\bold{A}</math>.
  • The non-zero singular values of M (found on the diagonal entries of Σ) are the square roots of the non-zero eigenvalues of both <math>\bold{A}^T\bold{A}</math> and <math>\bold{A}\bold{A}^T</math>.


Applications

Applications that employ the SVD include computing the pseudoinverse, least squares fitting of data, multivariable control, matrix approximation, and determining the rank, range and null space of a matrix.


LU Decomposition