Singular Values Calculator
Calculate singular values of any matrix using eigenvalue decomposition of ATA
Matrix Input
Enter Matrix Elements
Singular Values
Computed Singular Values (σ)
Matrix Properties
ATA Matrix
Eigenvalues of ATA: 26.1803, 3.8197
Singular values: σᵢ = √(λᵢ) where λᵢ are eigenvalues of ATA
Example Calculation
Sample Matrix A
Consider the 2×2 matrix:
[1 4]
Calculation Steps
1. Calculate ATA = [10 14; 14 20]
2. Find eigenvalues of ATA: λ₁ ≈ 30.77, λ₂ ≈ -0.77
3. Singular values: σ₁ = √30.77 ≈ 5.55, σ₂ = √0.77 ≈ 0.88
4. The largest singular value gives the operator norm
Singular Values Properties
Non-negative
σᵢ ≥ 0
Always real and positive
Universal
Any matrix has them
Square or rectangular
Operator Norm
σ₁ = ||A||₂
Largest singular value
Singular Values vs Eigenvalues
Singular values exist for any matrix
Always real and non-negative
Eigenvalues only for square matrices
For symmetric matrices: σᵢ = |λᵢ|
For diagonal matrices: σᵢ = |Aᵢᵢ|
Understanding Singular Values
What are Singular Values?
Singular values are the square roots of the eigenvalues of ATA, where AT is the transpose of matrix A. They provide important information about the geometric properties of linear transformations represented by the matrix.
Key Properties
- •Non-negativity: σᵢ ≥ 0 for all i
- •Ordering: σ₁ ≥ σ₂ ≥ ... ≥ σₙ ≥ 0
- •Matrix rank: Number of non-zero singular values
- •Operator norm: ||A||₂ = σ₁
Calculation Method
Step 1: Calculate AT (transpose of A)
Step 2: Compute ATA (matrix multiplication)
Step 3: Find eigenvalues λᵢ of ATA
Step 4: Singular values: σᵢ = √λᵢ
Applications
- •Image compression and processing
- •Data analysis and dimensionality reduction
- •Numerical analysis and condition numbers
- •Signal processing and filtering