SVD Calculator
Calculate complete Singular Value Decomposition A = UΣVT
Matrix Input
Enter Matrix A
SVD Decomposition: A = UΣVT
Singular Values
Matrix U (2×2)
Matrix Σ (2×2)
Matrix VT (2×2)
Verification: UΣVT
This should match your original matrix A (within numerical precision)
Step-by-Step Example
Given Matrix A
Consider the 2×2 matrix:
[3 -5]
SVD Steps
Step 1: Calculate ATA
Step 2: Find eigenvalues of ATA
Step 3: Singular values σᵢ = √(eigenvalues)
Step 4: Form Σ matrix with singular values
Step 5: Calculate V from eigenvectors of ATA
Step 6: Calculate U from A·V = U·Σ
SVD Properties
Orthogonal Matrix
UTU = I
Left singular vectors
Diagonal Matrix
σ₁ ≥ σ₂ ≥ ... ≥ 0
Singular values
Orthogonal Matrix
VVT = I
Right singular vectors
SVD Applications
Data compression and noise reduction
Principal Component Analysis (PCA)
Least squares solutions
Image processing and recognition
Pseudoinverse computation
Recommendation systems
Understanding Singular Value Decomposition
What is SVD?
Singular Value Decomposition (SVD) is a fundamental matrix factorization that decomposes any real matrix A into the product of three matrices: A = UΣVT, where U and V are orthogonal matrices and Σ is diagonal.
Mathematical Foundation
- •A: Original m×n matrix
- •U: m×m orthogonal matrix (left singular vectors)
- •Σ: m×n diagonal matrix (singular values)
- •VT: n×n orthogonal matrix (right singular vectors)
Calculation Algorithm
Step 1: Compute ATA
Step 2: Find eigenvalues and eigenvectors of ATA
Step 3: Singular values σᵢ = √(eigenvalues)
Step 4: V matrix from eigenvectors of ATA
Step 5: U matrix from A·V = U·Σ
Key Properties
- •Exists for any real matrix (rectangular or square)
- •Singular values are non-negative and ordered
- •Optimal low-rank approximation
- •Reveals matrix rank and null space