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Where:
 
Where:
 
* '''A''' is an <math>m \times n</math> real or complex matrix
 
* '''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
+
* '''U''' is an <math>m \times m</math> real or complex [https://en.wikipedia.org/wiki/Unitary_matrix unitary] '''orthonormal''' matrix
 
* <math>\bold{\Sigma}</math> is an <math>m \times n</math> 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 <math>n \times n</math> real or complex unitary matrix.
+
* '''V''' is an <math>n \times n</math> real or complex unitary '''orthonormal''' 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.
+
'''NOTES:'''
 +
* The diagonal entries <math>\sigma_i</math> of <math>\bold{\Sigma}</math> are known as the singular values of '''A''' and can be viewed as the semiaxes of an n-dimensional ellipsoid.
 +
* The columns of '''U''' and the columns of '''V''' are called the left-singular vectors and right-singular vectors of '''A''', respectively.
  
  
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=== LU Decomposition ===
 
=== LU Decomposition ===
 +
LU Decomposition, or LU Factorization is the process of decomposing a matrix A this way:
 +
<math>\left [ \bold{A} \right ] = \left [ \bold{L} \right ] \cdot \left [ \bold{U} \right ]</math>
 +
 +
Where:
 +
* '''L''' is a '''L'''ower triangular matrix
 +
* '''U''' is an '''U'''pper triangular matrix
 +
 +
 +
<math>
 +
\begin{bmatrix}
 +
          a_{11} & a_{12} & a_{13} \\
 +
          a_{21} & a_{22} & a_{23} \\
 +
          a_{31} & a_{32} & a_{33}
 +
        \end{bmatrix} =
 +
      \begin{bmatrix}
 +
          l_{11} & 0 & 0 \\
 +
          l_{21} & l_{22} & 0 \\
 +
          l_{31} & l_{32} & l_{33}
 +
        \end{bmatrix}
 +
        \begin{bmatrix}
 +
          u_{11} & u_{12} & u_{13} \\
 +
          0 & u_{22} & u_{23} \\
 +
          0 & 0 & u_{33}
 +
        \end{bmatrix}.
 +
</math>
 +
 +
 +
==== Applications ====
 +
 +
Computers usually ''solve square systems of linear equations'' using the LU decomposition, and it is also a key step when ''inverting a matrix'', or ''computing the determinant of a matrix''.

Revision as of 17:34, 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 orthonormal 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 orthonormal matrix.


NOTES:

  • The diagonal entries <math>\sigma_i</math> of <math>\bold{\Sigma}</math> are known as the singular values of A and can be viewed as the semiaxes of an n-dimensional ellipsoid.
  • 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 A are a set of orthonormal eigenvectors of <math>\bold{A}\bold{A}^T</math>.
  • The right-singular vectors of A are a set of orthonormal eigenvectors of <math>\bold{A}^T\bold{A}</math>.
  • The non-zero singular values of A (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

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

Where:

  • L is a Lower triangular matrix
  • U is an Upper triangular matrix


<math>

\begin{bmatrix}
          a_{11} & a_{12} & a_{13} \\
          a_{21} & a_{22} & a_{23} \\
          a_{31} & a_{32} & a_{33}
       \end{bmatrix} =
     \begin{bmatrix}
          l_{11} & 0 & 0 \\
          l_{21} & l_{22} & 0 \\
          l_{31} & l_{32} & l_{33}
       \end{bmatrix}
       \begin{bmatrix}
          u_{11} & u_{12} & u_{13} \\
          0 & u_{22} & u_{23} \\
          0 & 0 & u_{33}
       \end{bmatrix}.

</math>


Applications

Computers usually solve square systems of linear equations using the LU decomposition, and it is also a key step when inverting a matrix, or computing the determinant of a matrix.