I'm reading a book about inverse analysis and trying to figure out how the authors do the inversion.
Assume that matrix $C$ is
$$
C
~=~
\begin{bmatrix}
88.53 & -33.60 & -5.33 \\
-33.60 & 15.44 & 2.67 \\
-5.33 & 2.67 & 0.48
\end{bmatrix}
$$
and at some point authors diagonalize this matrix to calculate matrix $P$ using
$$
C^{-1}
~=~
P^{\rm t} L P
$$
where $L$ is a diagonal matrix of positive eigenvalues and the columns of $P$ are orthonormal eigenvectors.
The above equation for diagonalizing inverse of $C$ is a bit different from what is usually used and therefore I cannot calculate $P$ correctly (same as the book!).
So, that would be great if somebody can show me the way to calculate $P$.
$$
P
~=~
\begin{bmatrix}
0.93 & 0.36 & -0.03 \\
-0.36 & 0.90 & -0.23 \\
-0.06 & 0.23 & 0.97
\end{bmatrix}
$$
Best Answer
Have you heard of Jordan Normal Form
For your matrix:
$$C=\left(\begin{matrix} 88.53 & -33.60 &-5.33\\ -33.60 & 15.44 & 2.67\\ -5.33 & 2.67 & 0.48 \end{matrix}\right)$$
We would find the eigenvalues and the eigenvectors and then diagonalize it such that:
$$C = P J P^{-1}$$
For this matrix, we have:
$\lambda_1 = 101.976, v_1 = (0.929997, -0.362899, -0.0583849)$
$\lambda_2 = 2.47102 , v_2 = (0.366307, 0.901908, 0.228868)$
$\lambda_3 = 0.00312608 , v_3 = (-0.0303981, -0.234233, 0.971705)$
Using these values, we can now diagonalize the matrix $C$.
$$C=\left(\begin{matrix} 88.53 & -33.60 &-5.33\\ -33.60 & 15.44 & 2.67\\ -5.33 & 2.67 & 0.48 \end{matrix}\right) = P J P^{-1}$$
Where: $$P = \left(\begin{matrix} 0.929997 & 0.366307 & -0.0303981 \\ -0.362899 & 0.901908 & -0.234233 \\ -0.0583849 & 0.228868 & 0.971705 \end{matrix}\right)$$
$$J = \left(\begin{matrix} 101.976 & 0 & 0 \\ 0 & 2.47102 & 0 \\ 0 & 0 & 0.00312608 \end{matrix}\right)$$
$$P^{-1} = \left(\begin{matrix} 0.929997 & -0.362899 & -0.0583849 \\ 0.366307 & 0.901908 & 0.228868 \\ -0.0303981 & -0.234233 & 0.971705 \end{matrix}\right)$$
Notice that the columns of $P$ are from the linear combination of the eigenvectors $[v_1 | v_2 | v_3]$.
Notice that $J$ is the diagonal of the eigenvalues $\lambda_1, \lambda_2, \lambda_3$.
Note the $P^{-1}$ is just the inverse of $P$.
Lastly, you should understand that this works only when a matrix is diagonalizable (see site referenced above).
Regards