1. What Are Singular Values?

The singular values (σ_i) of a matrix are the square roots of the non-negative eigenvalues of \(A^TA\) (or \(AA^T\)). They provide important information about the matrix's properties and are fundamental in singular value decomposition (SVD).

2. How Are Singular Values Calculated?

The calculator uses the following mathematical relationship:

Where:

• \( \sigma_i \) — Singular values of matrix A
• \( \lambda_i \) — Eigenvalues of \( A^TA \) (or \( AA^T \))
• A — Input matrix (m × n)

Explanation: The singular values are always non-negative real numbers, ordered from largest to smallest.

3. Importance of Singular Values

Details: Singular values are crucial in many applications including matrix approximations, principal component analysis (PCA), signal processing, and solving least squares problems.

4. Using the Calculator

Tips: Enter your matrix using comma separated values within rows and semicolon separated 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: For square symmetric matrices, singular values are absolute values of eigenvalues. For general matrices, they're related via \(A^TA\) or \(AA^T\).

Q2: How many singular values does a matrix have?
A: An m×n matrix has min(m,n) singular values.

Q3: What does a zero singular value indicate?
A: It indicates linear dependence in the matrix columns/rows and that the matrix is rank-deficient.

Q4: Are singular values always real numbers?
A: Yes, singular values are always non-negative real numbers, even for complex matrices.

Q5: What's the condition number of a matrix?
A: It's the ratio of largest to smallest singular value, measuring numerical stability.