Manual SVD Calculation

Singular Value Decomposition (SVD)

SVD can be performed on any rectangular or square matrix.

In SVD, U and V are unitary matrices (orthogonal if the matrix is real), satisfying the conditions UU^H = I and VV^H = I.

Computing the condition number is often important—it is defined as the ratio of the largest singular value to the smallest non-zero singular value in the diagonal matrix of singular values. A high condition number indicates a nearly singular or ill-conditioned matrix.

 

For a Matrix,

 

Step 1: We normalize each column

We get, H=

We divided the elements of the first column by √(2² + 3²) = √13, and proceeded similarly for the other columns.

Here singular values are not in decreasing order.

Step 2: Now we arrange the singular values in decreasing order

H=

 

That implies,

H = UΣV^H

Again assume, the first matrix is U (unitary matrix), the middle one is Σ (eigenmatrix), and 3rd matrix is V (unitary matrix).

Alternatively, UU^H=I,     V^HV=VV^H=I

Σ =

In the above matrix, σ₁=√52, σ₂=√13, σ₃=2, and Singular values are in decreasing order.

LU Decomposition using the Doolittle Method (with Partial Pivoting)

In this method, a pivoting matrix is used—typically an identity matrix that is modified to rearrange the rows such that the largest element in each column is moved to the diagonal position.

L represents the lower triangular matrix, and U represents the upper triangular matrix.

Using partial pivoting, LU decomposition can be performed on any square matrix to enhance numerical stability. 

Further Reading

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