"SIMILARITY TRANSFORMATIONS"
A matrix can be transformed to another one (which is diagonal) by means of a similitarity transformation
A given "A" matrix will be transformed to a  new one
"A' " will be diagonal
A diagonal matrix has zero all its elements except the values in the diagonal
The diagonal elements are the eigenvalues
We have a "B" matrix
"B" is a 3x3 matrix
The no null elements of "B" will be located on the diagonal
Such values in the matrix diagonal correspond to the eigenvalues of the matrix
We write the transformed "B' " matrix
This will be a very important tool
We need to diagonalize matrices to make predictions in quantum mechanics
The eigenvalues will be the possible results if a measurement is performed
These transformations will be usefull to diagonalize matrices and to obtain the eigenvalues which will be the results of the measurements
For example, we apply such a process to the "C" matrix
"C" is the original matrix and "D" is the diagonal matrix (resulting matrix)
We diagonalize the original "C" matrix
Then we calculate the "C" eigenvectors or eigenfunctions
We normalize the eigenfunctions
Then we construct the "S" matrix where the eigenvectors of the "C" matrix are the rows of "S"
If "S" is unitary, "S^+=S^-1"
An Unitary matrix transform from an orthonormal basis to an orthonormal basis
Now we repeat again the process
"C"is the original matrix. Then we obtain the eigenvectors of "C"
"C" original matrix
Then, obtain eigenvalues
Then construct "S"
The rows are the eigenvectors
eigenvectors from "C", the original matrix
In this way we obtain "D" that is the result of a similarity transformation
In general a "S" matrix will be unitary
To diagonalize Hamiltonians will be very usefull
