Matrices

Matrix Decomposition Methods

QR Decomposition

QR Decomposition, or QR Factorization is the process of decomposing a square 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 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 can be viewed as the matrix form of Gaussian elimination. It is the process of decomposing a square 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.