mM = {{-7, -16, 4}, {6, 13, -2}, {12, 16, 1}}

Eigensystem[mM]

mP = Transpose[{{1, 0, 3}, {-4, 3, 0}, {-2, 1, 2}}]

?Diagonal*

DiagonalMatrix[{5, 5, -3}]

?Inverse*

mM - mP . DiagonalMatrix[{5, 5, -3}] . Inverse[mP]

mP

mM . mP