Matrices

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>

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.