Matrix Decomposition Methods

QR Decomposition

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

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:

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

Where:

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

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