Question

Singular value decomposition of a matrix

Answer 1

answer is : In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics.

Formally, the singular-value decomposition of an {\displaystyle m\times n} real or complex matrix {\displaystyle \mathbf {M} } is a factorization of the form {\displaystyle \mathbf {U\Sigma V^{*}} }, where {\displaystyle \mathbf {U} } is an {\displaystyle m\times m} real or complex unitary matrix, {\displaystyle \mathbf {\Sigma } }is a {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and {\displaystyle \mathbf {V} } is an {\displaystyle n\times n} real or complex unitary matrix. The diagonal entries {\displaystyle \sigma _{i}} of {\displaystyle \mathbf {\Sigma } } are known as the singular values of {\displaystyle \mathbf {M} }. The columns of {\displaystyle \mathbf {U} } and the columns of {\displaystyle \mathbf {V} } are called the left-singular vectors and right-singular vectors of {\displaystyle \mathbf {M} }, 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 MM∗.The right-singular vectors of M are a set of orthonormal eigenvectors of M∗M.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 M∗M and MM∗.

Applications that employ the SVD include computing the pseudoinverse, least squaresfitting of data, multivariable control, matrix approximation, and determining the rank, range and null space of a matrix.

Answer 2

Answer:

The singular value decomposition of a matrix A is the factorization of A into the product of three matrices A = UDV T where the columns of U and V are orthonormal and the matrix D is diagonal with positive real entries. ... The columns of U are called the left singular vectors and they also form an orthogonal set.