Singular Value Decomposition factorizes any matrix A into U Σ V^T, where U and V are orthogonal and Σ is diagonal with singular values.

How to compute SVD step by step?

1) Compute A^T A, 2) find eigenvalues & eigenvectors, 3) singular values = sqrt(eigenvalues), 4) V = eigenvectors, 5) U = A V Σ^{-1}

🧮 SVD Calculator

Step-by-step singular value decomposition · up to 5×5 · color-coded matrices

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Rows: Cols:

💡 Steps: A^TA → eigenvalues → singular values → V → U

(waiting for input)

σ (singular values) —

👆 Click "Compute SVD" to see the full decomposition.

Choose rank k: 2

          A ≈ Uₖ Σₖ Vₖᵀ
        

Frobenius error: —

📈 PCA connection: Principal components are right singular vectors (V). Singular values ∝ variance explained.

🧠 LSA / topic modeling: SVD of term‑document matrix yields latent topics (UΣ = document‑topic, V = term‑topic).

What is SVD? Factorizes any real matrix A into U Σ Vᵀ, with orthogonal U, V and diagonal Σ.

How to interpret singular values? They are the square roots of eigenvalues of AᵀA, sorted descending. Indicate importance of each singular vector.

Why SVD for PCA? PCA uses eigenvectors of covariance (AᵀA). SVD gives them directly as V, and singular values relate to explained variance.

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