The eigenvalues of a Hermitian matrix are alla)realb)imaginaryc)of modulus oned)real and positiveCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Explanation:

A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. It means that it satisfies the condition A = A* where A* is the conjugate transpose of matrix A.

Real Eigenvalues

Let λ be an eigenvalue of a Hermitian matrix A, then there exists a non-zero vector x such that Ax=λx. Taking the conjugate transpose of both sides, we get

(Ax)* = (λx)*

x*A* = λ*x*

Multiplying both sides by vector x, we get

x*A*x = λ*x*x*

Here, x*x* is a non-negative real number. As A is Hermitian, x*A*x is also a real number. Hence the eigenvalues of a Hermitian matrix are real.

Positive Eigenvalues

Let A be a positive definite Hermitian matrix, then for all non-zero vectors x, we have x*A*x > 0. Hence, all the eigenvalues of A are positive.

Conclusion

Therefore, the eigenvalues of a Hermitian matrix are always real.

Answer

Real and postive

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