K  =  {{2, -1}, {-1, 2}}

MatrixForm[K]
Eigensystem[K]

Q  =  1/Sqrt[2] Eigenvectors[K]

L  =  DiagonalMatrix[Eigenvalues[K]]
MatrixForm[Q]
MatrixForm[L]

MatrixForm[Transpose[Q]  .  (L . Q)]