Eigenvalues and diagonalizable matrix Please show work Thank

Eigenvalues and diagonalizable matrix. Please show work. Thank you.

Prove that if X M_n(R) is a diagonalizable matrix (with real eigenvalues), then the matrix Y = X + X^-1 - 2I_2 also has positive eigenvalues?

Solution

A square matrix A is said to be diagonalizable if A is similar to a diagonal matrix, i.e. if A PDP¹where P is invertible and D is a diagonal matrix. Where P is called the model matrix.

This property (that the eigenvalues of a diagonal matrix coincide with its diagonal entries and the eigenvectors corresponds to the corresponding coordinate vectors)is so useful and important that in practice one often tries to make a change of coordinates just so that this will happen. Unfortunately, this is not always possible; however, if it is possible to make a change of coordinates so that a matrix becomes diagonal we say that the matrix is diagonalizable. More formally,X is diagnalizable matrix then X^-1 is also dianlizable and Identity matrix is is also dianalizable.So Y=X+X^-1-2I₂ has positive eigen values.