Question

Let \( A \) be an \( n \times n \) real matrix. Consider the following statements.
(I) If \( A \) is symmetric, then there exists \( c \geq 0 \) such that \( A + c I_n \) is symmetric and positive definite, where \( I_n \) is the \( n \times n \) identity matrix.
(II) If \( A \) is symmetric and positive definite, then there exists a symmetric and positive definite matrix \( B \) such that \( A = B^2 \).
\text{Which of the above statements is/are true?}

• A. Only (I)
• B. Only (II)
• C. Both (I) and (II)
• D. Neither (I) nor (II)

Hint:

- Symmetric matrices can always be made positive definite by adding a scalar multiple of the identity matrix.
- Positive definite matrices have a unique square root decomposition.

Answer 1

The Correct Option is C

1) Statement (I):
This statement is true. If \( A \) is symmetric, we can add a scalar multiple of the identity matrix \( c I_n \) to make the matrix \( A + c I_n \) positive definite. Since the eigenvalues of a symmetric matrix are real, adding a positive scalar to the diagonal entries ensures that the matrix becomes positive definite.
2) Statement (II):
This statement is also true. If \( A \) is symmetric and positive definite, it can be decomposed as \( A = B^2 \), where \( B \) is a symmetric and positive definite matrix. This follows from the spectral decomposition theorem, which allows the square root decomposition of positive definite matrices.
The correct answer is (C) Both (I) and (II).