Question

Let \( M \) be a \( 3 \times 2 \) real matrix having a singular value decomposition as \( M = U S V^T \), where the matrix \( S = \begin{bmatrix} \sqrt{3} & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}^T \), \( U \) is a \( 3 \times 3 \) orthogonal matrix, and \( V \) is a \( 2 \times 2 \) orthogonal matrix. Then which of the following statements is/are true?

• A. The rank of the matrix \( M \) is 1
• B. The trace of the matrix \( M^T M \) is 4
• C. The largest singular value of the matrix \( (M^T M)^{-1} M^T \) is 1
• D. The nullity of the matrix \( M \) is 1

Hint:

In Singular Value Decomposition (SVD), the rank of a matrix equals the number of non-zero singular values, the trace of \( M^T M \) equals the sum of the squares of singular values, and the inverse-based transformations are linked to reciprocal singular values.

Answer 1

The Correct Option is B, C

Step 1: Understanding the given information.
We are told that the matrix \( M \) is \( 3 \times 2 \) and has a singular value decomposition (SVD): \[ M = U S V^T, \] where \( U \) is a \( 3 \times 3 \) orthogonal matrix, \( V \) is a \( 2 \times 2 \) orthogonal matrix, and \( S \) is a \( 3 \times 2 \) diagonal-like matrix that contains the singular values of \( M \). The given \( S \) matrix is: \[ S = \begin{bmatrix} \sqrt{3} & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}. \] From this, we can directly read the singular values of \( M \): \[ \sigma_1 = \sqrt{3}, \quad \sigma_2 = 1. \]
Step 2: Finding the rank of \( M \).
The rank of a matrix is equal to the number of non-zero singular values.
Since both \( \sqrt{3} \) and \( 1 \) are non-zero, the rank of \( M \) is 2.
Therefore, statement (A) “The rank of the matrix \( M \) is 1” is false.

Step 3: Finding the trace of \( M^T M \).
We know that \( M^T M = V S^T S V^T \).
Because \( V \) is an orthogonal matrix, the trace of \( M^T M \) equals the trace of \( S^T S \).
Now, \[ S^T S = \begin{bmatrix} \sqrt{3} & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} \sqrt{3} & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 3 & 0 \\ 0 & 1 \end{bmatrix}. \] Hence, \[ \text{trace}(M^T M) = 3 + 1 = 4. \] So, statement (B) “The trace of \( M^T M \) is 4” is true.

Step 4: Checking the largest singular value of \( (M^T M)^{-1 M^T \).}
The singular values of \( M \) are \( \sigma_1 = \sqrt{3} \) and \( \sigma_2 = 1 \).
The matrix \( M^T M \) will have eigenvalues \( \sigma_1^2 = 3 \) and \( \sigma_2^2 = 1 \).
Thus, \[ (M^T M)^{-1} = \begin{bmatrix} 1/3 & 0 \\ 0 & 1 \end{bmatrix}. \] Now, consider the matrix \( (M^T M)^{-1} M^T \). The singular values of this matrix are the reciprocals of the singular values of \( M \). Hence, the largest singular value of \( (M^T M)^{-1} M^T \) is: \[ \frac{1}{\sigma_2} = \frac{1}{1} = 1. \] Therefore, statement (C) is true.

Step 5: Finding the nullity of \( M \).
The nullity of a matrix is given by: \[ \text{Nullity}(M) = \text{Number of columns} - \text{Rank}(M). \] Since \( M \) has 2 columns and rank 2, \[ \text{Nullity}(M) = 2 - 2 = 0. \] Therefore, statement (D) “The nullity of the matrix \( M \) is 1” is false.

Step 6: Conclusion.
Hence, the correct statements are (B) and (C).