Singular value decomposition (svd) in MATLAB - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is Singular Value Decomposition (SVD) in MATLAB?

SVD is a method to break down a matrix into three simpler matrices: U, S, and V, where U and V are orthogonal, and S is diagonal with singular values.

beginner

What does the MATLAB command [U,S,V] = svd(A) return?

It returns matrices U, S, and V such that A = U * S * V'. U and V are orthogonal matrices, and S is a diagonal matrix with singular values.

intermediate

Why are singular values in matrix S always non-negative?

Because singular values represent the lengths of the semi-axes of the ellipse formed by the matrix transformation, which cannot be negative.

intermediate

How can SVD help in image compression in MATLAB?

By keeping only the largest singular values and corresponding vectors, you can approximate the image with fewer data, reducing size while keeping important features.

intermediate

What is the shape of matrices U, S, and V when A is m-by-n?

U is m-by-m, S is m-by-n diagonal, and V is n-by-n. This ensures the multiplication U*S*V' reconstructs A.

In MATLAB, what does the matrix S contain after [U,S,V] = svd(A)?

A Random values

B Eigenvalues of A

C Inverse of A

D Singular values on the diagonal

Matrix S contains the singular values of A on its diagonal, sorted from largest to smallest.

Which MATLAB command reconstructs the original matrix A from its SVD components?

A U * S * V'

B U + S + V

C S * U * V

D V * S * U'

The original matrix A is reconstructed by multiplying U, S, and the transpose of V: A = U * S * V'.

What property do matrices U and V have in SVD?

A They are zero matrices

B They are diagonal matrices

C They are orthogonal matrices

D They are identity matrices

U and V are orthogonal matrices, meaning their columns are perpendicular unit vectors.

If A is 4-by-3, what is the size of matrix V in [U,S,V] = svd(A)?

A 3-by-4

B 3-by-3

C 4-by-3

D 4-by-4

Matrix V is n-by-n, so for A of size 4-by-3, V is 3-by-3.

Why might you use only the first few singular values in SVD?

A To approximate the matrix with less data

B To make the matrix larger

C To invert the matrix

D To find eigenvalues

Using only the largest singular values helps approximate the matrix while reducing data size, useful in compression.

Explain how Singular Value Decomposition breaks down a matrix and what each component represents.

Think of U and V as rotations and S as scaling.

Describe a practical use of SVD in MATLAB, such as image compression, and how it works.

Focus on how fewer singular values can still keep the main picture.