Prove that if X MnR is a diagonalizable matrix with real ei

Prove that if X ? Mn(R) is a diagonalizable matrix (with real eigenvalues), then the matrix Y = X + X?1 ? 2In also has positive eigenvalues?

No copy and paste please from other answers... original procedure recommended...

Prove that if X M( matrix Y-X +X-1-2I R) is a diagonalizable matrix with real eigenvalues), then the t y le n also has positive eigenvalues!

Solution

The inverse for a diagonal matrix is obtained by replacing each element in the diagonal with its reciprocal.

If X is diagonalizable then there exists a matrix P such that P^-1XP is a diagonal matrix with eigenvalues as the diagonal elements.

Given matrix Y is

Y = X + X^-1 - 2I_n

multiplying it with and P and P^-1

P^-1YP = P^-1( X + X^-1 - 2I_n ) P

= P^-1XP + P^-1X^-1P - 2P^-1I_nP

As mentioned above P^-1XP will be a diagonal matrix with diagonal elements a1,a2...an

P^-1X^-1P = ( P^-1XP )^-1

So as mentioned above this will be a diagonal matrix with diagonal elements 1/a1, 1/a2, .... 1/an.

2P^-1I_nP = 2I_n

P^-1YP will be a diagonal matrix with ( a_n + 1/a_n - 2 ) as the diagonal elements.

As mentioned in the second point as P^-1YP is a diagonal matrix the diagonal elements will be the eigenvalues of Y

We know that x + 1/x >= 2

So the eigenvalues of Y will always be positive.