Linear Algebra: UNIT IV: Matrix Decomposition
Singular Value Decomposition
Matrix Decomposition
A singular value decomposition (SVD) is a generalization of this where A is an m×n which does not have to be symmetric or even square.
SINGULAR VALUE
DECOMPOSITION
If
A is a symmetric n×n matrix, then A has real eigenvalues λ1, λ2,
λ3 ... ... λn (possibly repeated) and Rn has
an orthonormal basis v1, v2, v3 …. vn
where each vector vi is an eigenvector of A with eigenvalue λi.
Then A = PDP‒1 where P is the matrix whose columns are v1,
v2, v3 ... vn and D is the diagonal matrix
whose diagonal entries are λ1, λ2, ... λn.
Since the vectors v1, v2, v3 ... vn
are orthonormal, the matrix P is orthogonal. PTP = I, so we can
alternately write this equation as
A = PDPT
A
singular value decomposition (SVD) is a generalization of this where A is an m×n
which does not have to be symmetric or even square.
1. Singular values
Let
A be an m×n matrix. Before explaining the singular value decomposition, we need
to define the singular values of A.
Consider
the matrix ATA. This is a symmetric n×n matrix and its eigenvalues
are real.
If
λ is an eigenvalue of ATA; then λ>0.
(ie)
Let X be an eigenvector of ATA with eigenvalue λ.
||AX||2 = (AX) • (AX)
=
(AX)T(AX)
=
XTATAX
=
XT (λX)
=
λ XTX
|| AX ||2 = λ|| X ||2
Since
|| AX ||2 ≥ 0, it follows from the above equation λ||X||2≥0.
Since || X||2>0, we deduce that λ≥0.
Let
λ1, λ2, λ3 ... λn be the
eigenvalues of ATA, with repetitions. Order these so that λ1
≥ λ2 ≥ λ3 …. ≥ λn
≥0.
Let
σ√(λi), so that σ1 ≥ σ2 ≥ σ3 ... ≥ σn ≥ 0.
The
numbers σ1 ≥ σ2 ≥ σ3 ... ≥ σn ≥ 0 are called the singular
values of A.
2. Singular value decomposition
Let
A be an m×n matrix with singular values σ1 ≥ σ2 ≥ σ3
... ≥ σn ≥ 0. Let r denote
the number of non−zero singular values of A or equivalently the rank of A.
A
singular value decomposition of A is a factorization
A = UΣVT
Here
U is an m×m orthogonal matrix and the column of U are the left−singular vectors
of A.
V
is an n×n orthogonal matrix, and the columns of V are the right singular
vectors of A. Σ is an m×n rectangular diagonal matrix whose ith diagonal
entry equals the ith singular value σi for i = 1, 2... r.
All
other entries of Σ are zero. The diagonal entries of Σ are the singular values,
which are non−negative and typically arranged in descending order.
Linear Algebra: UNIT IV: Matrix Decomposition : Tag: maths, mathematics : Matrix Decomposition - Singular Value Decomposition
Linear Algebra: UNIT IV: Matrix Decomposition
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Linear Algebra
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