Spectral Theorem
Every real, symmetric matrix is orthogonally diagonalizable.
Theorem 1 (Spectral Theorem) Every real symmetric matrix is diagonalizable.
Let
- The eigenvalues of
are real. - There exists an orthonormal basis
for consisting of the eigenvectors of . That is, there is an orthogonal matrix so that .
The term spectrum refers to the eigenvalues of a matrix, or more, generally a linear operator. In Physics, the spectral energy lines of atoms (e.g. Balmer lines of the Hydrogen atom), are characterized as the eigenvalues of the governing quantum mechanical Schrodinger operator.
Proof.
Claim. The eigenvalues of
Since, for a symmetric matrix
Or using the dot-product notation, we could write:
Suppose
We can now take the complex conjugate of the eigenvalue equation. Remember that
Using the eigenvalue equation (Equation 2), we can write:
Alternatively, using Equation 1 and Equation 3, we have:
Consequently,
Since,
Claim.
We proceed by induction.
For
Inductive hypotheis. Every
Claim. Let
Every square matrix
Now, we can extend this to a basis
Now, we huddle these basis vectors together as column-vectors of a matrix and formulate the matrix
By definition,
Define
Step I.
We have:
We are now going to try and write
Step II. The structure of
The way we do this, is to consider the matrix
Now, remember that
This is the first column of the matrix
So, we can write the matrix
The first row and the first column are satisying the need to be diagonal.
Step III.
We know that
Now, define the matrix
Claim. Our claim is that
- We have:
But,
So,
Thus,
- Well, let’s compute
.
Since