Question:
Let \( A \in M_n({C}) \) be a normal matrix. Consider the following statements:
1. If all the eigenvalues of \( A \) are real, then \( A \) is Hermitian.
2. If all the eigenvalues of \( A \) have absolute value 1, then \( A \) is unitary.
Which one of the following is correct?
Let \( A \in M_n({C}) \) be a normal matrix. Consider the following statements:
1. If all the eigenvalues of \( A \) are real, then \( A \) is Hermitian.
2. If all the eigenvalues of \( A \) have absolute value 1, then \( A \) is unitary.
Which one of the following is correct?
1. If all the eigenvalues of \( A \) are real, then \( A \) is Hermitian.
2. If all the eigenvalues of \( A \) have absolute value 1, then \( A \) is unitary.
Which one of the following is correct?
Show Hint
For normal matrices, use the properties of eigenvalues to determine Hermitian or unitary nature.
Updated On: Feb 1, 2025
- Both I and II are TRUE
- I is TRUE and II is FALSE
- I is FALSE and II is TRUE
- Both I and II are FALSE
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Verifying the Hermitian property.
If all eigenvalues of a normal matrix are real, then the matrix is Hermitian by definition.
Step 2: Verifying the unitary property.
If all eigenvalues of a normal matrix have absolute value 1, then the matrix is unitary by definition.
Step 3: Conclusion.
Both statements are true. The correct answer is \( {(1)} \).
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