Singular Value Decomposition

Key idea

Factor any matrix as UΣVᵀ.

In depth

Every matrix factors as A = UΣVᵀ with orthogonal U, V and diagonal Σ of non-negative singular values. It describes A as rotate, stretch, rotate. It works for any shape matrix and underlies compression and PCA.

Worked example

A rank-1 approximation keeps only the largest singular value σ₁u₁v₁ᵀ.

Formula

Real-world uses

• Compressing images and recommending products
• Reducing noise in large datasets

Common mistake

Confusing singular values with eigenvalues; they are square roots of eigenvalues of AᵀA.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Linear Algebra

• Triple Product – Scalar triple product gives parallelepiped volume.
• Homogeneous Systems – Ax = 0 always has the trivial solution.
• Cofactor Expansion – Compute determinants recursively.
• Properties of Determinants – Multiplicative; row swaps change sign.
• Geometric Meaning of Determinants – Area or volume scaling factor.
• Cramer’s Rule for Systems – Solve systems using determinants.