Singular Value Decomposition

Author:Ku, Yin Bon (Albert)

As we know, diagonalization is a very useful technique of decomposing a matrix so that we can do computations more easily. However, there are two main limitation of diagonalization:

It only applies to square matrices.
Not every square matrix is diagonalizable.

But it turns out that there exists a matrix decomposition which is equally if not more useful than diagonalization that works for ANY matrix (even the non-square ones!). This is the so-called singular value decomposition (SVD): For any real

matrix

, there exists an

orthogonal matrix

and an

orthogonal matrix

such that

, where

is a diagonal

matrix. Moreover, the diagonal entries of

, denoted by

for the

row (

), are nonnegative and can be arranged in descending order. The positive

are called the singular values of

and the column vectors of

and

are called the left and right singular vectors of

respectively.

An Example

Let

. The SVD of

is as follows:

(Note: SVD is not unique.) This is an useful online SVD calculator.

Singular Value Decomposition

An Example

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