Singular Value Decomposition Calculator

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Example

Created on 2024-06-20Asked by Mason Williams (Solvelet student)

Compute the singular value decomposition (SVD) of the matrix

\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Solution

To compute the singular value decomposition (SVD) of the matrix

\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

: 1. **Define the matrix:**

A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.

2. **Compute the singular values:** The singular values are the square roots of the eigenvalues of

A^TA

A^TA = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}^T \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.

The eigenvalues are

1

and

1

, so the singular values are

1

and

1

. 3. **Compute the matrices

U

\Sigma

, and

V

:**

U = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad \Sigma = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad V = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.

4. **Result:** The SVD of the matrix is:

A = U \Sigma V^T = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}^T.

Solved on Solvelet with Basic AI Model

Some of the related questions asked by James Hall on Solvelet

1. Compute the singular value decomposition of the matrix

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

.2. Use singular value decomposition to approximate the matrix

A = \begin{pmatrix} 2 & 3 \\ 3 & 4 \end{pmatrix}

with a rank-1 matrix.,

Definition Singular Value Decomposition (SVD) is a factorization of a real or complex matrix. It extends the eigendecomposition of a square matrix to any m×n matrix. For example: LET A BE A MATRIX, THEN, SVD WILL BE A=UΣV∗., WHERE U AND V ARE UNITARY MATRICES AND Σ IS DIAGONAL MATRIX OF SINGULAR VALUES.

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