What is the nature of the diagonalizing matrix of certain types of matrices?

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Suppose I have a matrix, A of order n. I want to diagonalize it. To diagonalize, I will have to evaluate the eigenvalues of the matrix and the corresponding eigenvectors. Then I will have to form a matrix, P, using these eigenvectors. My question is what will be the nature of this matrix P when the nature of the initial matrix A has been specified to be real symmetric? Also, what will be the nature of P if A is Hermitian?