From the following statement, it seems matrix diagonalization is just eigen decomposition.
Diagonalizing a matrix is also equivalent to finding the matrix's eigenvalues, which turn out to be precisely the entries of the diagonalized matrix. Similarly, the eigenvectors make up the new set of axes corresponding to the diagonal matrix.
However, from what I have learned, Spectral Theorem is closest to this conclusion. But how the spectral theorem is related to it, or is there some other theorem grants this statement?
Spectral Theorem: Suppose that $V$ is a complex inner-product space and $T \in L(V)$. Then $V$ has an orthonormal basis consisting of eigenvectors of $T$ if and only if $T$ is normal.