Question

SVD can decompose a matrix A (of size m x n) into three components, namely U, V, and S, where U is an orthogonal matrix, S is a diagonal matrix, and V is an orthogonal matrix in such a way that A = U * S * VT. Read the below statements:

A) Dimension of U is m x r

B) S is a square matrix of dimension r x r

C) Dimension of VT is r x r

D) Dimension of VT is r x n
Here, r is the rank of matrix A. Which of the above statements are correct?

Answer 1

The dimension of U is m x r, S is a square matrix of dimension r x r, and the dimension of VT is r x n.

Step-by-step explanation:

The correct statements are:

• Statement A: The dimension of U is m x r. This is correct because U is an m x r matrix.
• Statement B: S is a square matrix of dimension r x r. This is correct because S is a diagonal matrix and therefore has the same number of rows and columns.
• Statement C: The dimension of VT is r x r. This is incorrect. The dimension of VT is n x r, as V is the transpose of V and swaps the dimensions.
• Statement D: The dimension of VT is r x n. This is correct, as mentioned in the previous statement.