Unlike the simple case of a matrix, in infinite dimensional vector spaces, the spectrum may be continuous.

The quintessential example of that is the spectrum of the position operator in quantum mechanics, in which any real number is a possible eigenvalue, since the particle may be found in any position. The associated eigenvectors are the corresponding Dirac delta functions.

Eigendecomposition of a matrix

Every invertible matrix

M

can be written as:

M = Q D Q^{- 1}

(1)

where:

$D$ is a diagonal matrix containing the eigenvalues of $M$
columns of $Q$ are eigenvectors of $M$

Note therefore that this decomposition is unique up to swapping the order of eigenvectors. We could fix a canonical form by sorting eigenvectors from smallest to largest in the case of a real number.

Intuitively, Note that this is just the change of bases formula, and so:

$Q^{- 1}$ changes basis to align to the eigenvectors
$D$ multiplies eigenvectors simply by eigenvalues
$Q$ changes back to the original basis

Eigendecomposition of a real symmetric matrix

The general result from eigendecomposition of a matrix:

M = Q D Q^{- 1}

(1)

becomes:

M = O D O^{T}

(2)

where

O

is an orthogonal matrix, and therefore has

O^{- 1} = O^{T}

Sylvester's law of inertia

The main interest of this theorem is in classifying the indefinite orthogonal groups, which in turn is fundamental because the Lorentz group is an indefinite orthogonal groups, see: all indefinite orthogonal groups of matrices of equal metric signature are isomorphic.

It also tells us that a change of bases does not the alter the metric signature of a bilinear form, see matrix congruence can be seen as the change of basis of a bilinear form.

The theorem states that the number of 0, 1 and -1 in the metric signature is the same for two symmetric matrices that are congruent matrices.

For example, consider:

A = [2223]

(1)

The eigenvalues of

A

are

1

and

4

, and the associated eigenvectors are:

v_{1} = [- 2, 1]^{T} v_{4} = [2 / 2, 1]^{T}

(2)

symPy code:

A = Matrix([[2, sqrt(2)], [sqrt(2), 3]])
A.eigenvects()

and from the eigendecomposition of a real symmetric matrix we know that:

A = P D P^{T} = [- 2 1 2 / 2 1] [1004] [- 2 2 / 2 11]

(3)

Now, instead of

P

, we could use

P E

, where

E

is an arbitrary diagonal matrix of type:

[e_{1} 0 0 e_{2}]

(4)

With this, would reach a new matrix

B

B = (P E) D (P E)^{T} = P (E D E^{T}) P^{T} = P (E E D) P^{T}

(5)

Therefore, with this congruence, we are able to multiply the eigenvalues of

A

by any positive number

e_{1}^{2}

and

e_{2}^{2}

. Since we are multiplying by two arbitrary positive numbers, we cannot change the signs of the original eigenvalues, and so the metric signature is maintained, but respecting that any value can be reached.

Note that the matrix congruence relation looks a bit like the eigendecomposition of a matrix: