Matrices

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http://www.seas.ucla.edu/~vandenbe/133A/


Contents

Matrix Decomposition Methods

QR Decomposition

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

QR.png

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:

$ \left [ \bold{A} \right ] = \left [ \bold{U} \right ] \cdot \left [ \bold{\Sigma} \right ] \cdot \left [ \bold{V} \right ]^T $

SVD.png

Where:

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


NOTES:

  • The diagonal entries $ \sigma_i $ of $ \bold{\Sigma} $ 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 $ \bold{A}\bold{A}^T $.
  • The right-singular vectors of A are a set of orthonormal eigenvectors of $ \bold{A}^T\bold{A} $.
  • 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 $ \bold{A}^T\bold{A} $ and $ \bold{A}\bold{A}^T $.


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 can be viewed as the matrix form of Gaussian elimination. It is the process of decomposing a square matrix A this way: $ \left [ \bold{A} \right ] = \left [ \bold{L} \right ] \cdot \left [ \bold{U} \right ] $

LU.png

Where:

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


$ \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}. $


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.

Cholesky decomposition

Cholesky Decomposition => TODO

Applications

Useful for efficient numerical solutions and Monte Carlo simulations. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.