Question

show that any two eigenvectors corresponding to two distinct eigen values of a hermitian matrix are orthogonal​

Answer 1

Answer:

Take a real symmetric matrix M, and two distinct eigenvalues of M, λ1 and λ2, such that Mx1=λ1x1 and Mx2=λ2x2.

From Mx1=λ1x1, we get x2TMx1=λ1x2Tx1. From Mx2=λ2x2, we similarly obtain x1TMx2=λ2x1Tx2. But since M is symmetric, x2TMx1=x2TMTx1=(Mx2)Tx1=x1T(Mx2)=x1TMx2. Also, clearly x1Tx2=x2Tx1. Thus

λ1x1Tx2=λ2x1Tx2

(λ1−λ2)(x1Tx2)=0.

But λ1−λ2≠0. Hence x1Tx2=0, as required.