Question

How to check if something is an eigenvalue of a matrix?

Answer 1

If the square matrix is [A], then any eigenvalue, λ, satisfied the equation
[A] = λ[I]
where [I] is the identity matrix of the same size as [A].

To test whether λ is an eigenvalue, verify that
det([A] - λ[I]) = 0

Example:
Test whether 25.06 is an eigenvalue of the following matrix:

`[A]= \left[\begin{array}{ccc}15&0&-1\\0&25&1\\-1&1&9\end{array}\right]`
Test the determinant of [A] - 25.06[I] to see whether it is zero.

`det( \left[\begin{array}{ccc}15&0&-1\\0&25&1\\-1&1&9\end{array}\right] \, - 25.06 \left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] ) \\ =det \left[\begin{array}{ccc}-10.06&0&-1\\0&-0.06&1\\-1&1&-16.06\end{array}\right] \\ = 0.4262`

This determinant is not zero, but it is small compared with the diagonal elements of [A].
In practical terms (such as in engineering or physics) it is close to an eigenvalue of [A].
The actual eigenvalue is 25.0626