1. What Are Singular Values?

Singular values (\(\sigma_i\)) are the square roots of the eigenvalues of \(A^TA\) (or \(AA^T\)), where \(A\) is an m×n matrix. They are fundamental in singular value decomposition (SVD), a powerful matrix factorization method.

2. How Singular Values Are Calculated

The calculation process:

Where:

• \(\sigma_i\) — The i-th singular value
• \(\lambda_i\) — The i-th eigenvalue of \(A^TA\) (or \(AA^T\))
• \(A\) — The original matrix (m × n)

Calculation Steps:

1. Compute \(A^TA\) (or \(AA^T\))
2. Find the eigenvalues of this product matrix
3. Take the square roots of these eigenvalues
4. Order them in decreasing magnitude

3. Importance of Singular Values

Applications: Singular values are crucial in many areas including signal processing, statistics, and machine learning. They reveal important properties about the matrix such as its rank, stability, and sensitivity to perturbations.

4. Using the Calculator

Instructions: Enter your matrix in the text area using comma-separated values for columns and semicolon-separated values for rows. For example:
1, 2, 3; 4, 5, 6; 7, 8, 9 for a 3×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between singular values and eigenvalues?
A: Singular values are the square roots of eigenvalues of \(A^TA\) (or \(AA^T\)), not necessarily of A itself.

Q2: Are singular values always real and non-negative?
A: Yes, since \(A^TA\) is positive semidefinite, its eigenvalues (and thus their square roots) are always real and non-negative.

Q3: How many singular values does a matrix have?
A: An m×n matrix has min(m,n) singular values (some may be zero).

Q4: What does a zero singular value indicate?
A: A zero singular value indicates linear dependence in the columns/rows of the matrix and contributes to the null space.

Q5: What's the significance of the largest singular value?
A: The largest singular value (\(\sigma_1\)) represents the 2-norm of the matrix and its largest scaling factor in any direction.