Year 12s: what will be the first way you research your future options? >>

Eigenvalues/Eigenvectors

Consider the following four symmetric matrices

a)

\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}

\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}

\begin{pmatrix} 1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 1 \end{pmatrix}

\begin{pmatrix} 1 & 1&1 \\ 1 & 1&1\\1&1&1 \end{pmatrix}

QUESTION
Calculate the eigenvectors, where the eigenvalues are distinct, confirm that the eigenvectors are orthogonal. If the eigenvalues are degenerate, construct a suitable set of orthogonal vectors. (Btw, my first move is to calculate the eigenvalues. These matrices look suspicious. Is there a way to do this...quickly? All the eigenvalues should be real, why is this...?)

Any help would be appreciated, not sure where to start..

Reply 1

mik1a

The eigenvalues will be real for symmetric real matrices (can you prove this?).

As for the symmetry helping you, there's no general silver bullet but there are some 'tricks of the trade', most usefully the eigenvalues sum to the trace of the matrix (giving you the last one for free, if you find the rest by other means), and there are some block rules for calculating the determinant of a structured matrix (see link below) when you calculate the eigenvalues by the normal method (sub k from the diagonal):

https://en.m.wikipedia.org/wiki/Determinant#Block_matrices

Reply 2

atsruser

Original post by hezzlington

All the eigenvalues should be real, why is this...?

Any help would be appreciated, not sure where to start..

A real symmetric matrix is a special case of an Hermitian matrix, whose properties you can look up. The most important ones are that their eigenvalues are real, and their eigenvectors form an orthogonal set.

They crop up in quantum mechanics for this reason, where the matrix form of a wavefunction operator is an Hermitian matrix; the real eigenvalues correspond to the possible measured states of the operator (e.g. energy, angular momentum, etc) and the orthogonal eigenvectors correspond to the set of states that the system can be in after a measurement.

Reply 3

DFranklin

Original post by hezzlington

...Calculate the eigenvectors, where the eigenvalues are distinct, confirm that the eigenvectors are orthogonal. If the eigenvalues are degenerate, construct a suitable set of orthogonal vectors. (Btw, my first move is to calculate the eigenvalues. These matrices look suspicious. Is there a way to do this...quickly? All the eigenvalues should be real, why is this...?) In general, your best plan is going to be as you suggest: find the eigenvalues, then find the eigenvectors.

The other potential method is to directly find the eigenvectors; assuming you're supposed to know that "once you have n orthogonal eigenvectors, then you're done", it's potentially going to be quicker to find n orthogonal eigenvectors (since verifying that a vector v is an eigenvector is quick and easy), rather than do the (somewhat labourious) finding of each eigenvalue, then eigenvector.

Pulling off the second method is a lot easier with "nice" matrices such as the ones you have here. But to be honest, I still think there's very little chance that you'll be able to pull off the second method at this stage in your mathematical career however. But it's quite possible you'll be able to do it in a year or two.