The Characteristic Equation

The characteristic equation is simply the equation whose solutions are a matrix's eigenvalues. The name is older English for the same idea: eigenvalues used to be called "characteristic values" (the German "eigen" can also mean "characteristic"), so this is the equation for them. It's useful because guessing eigenvectors doesn't get you far; this gives you the eigenvalues first, and then each eigenvector comes from solving a null space. All we're really saying in $\det(A - \lambda I) = 0$ is: $A - \lambda I$ ($A$ with $\lambda$ subtracted from each diagonal entry), $\det$ (its determinant, which is zero exactly when the matrix squashes some nonzero vector to zero) and $= 0$ (find the $\lambda$ that make that happen). It looks heavy, but it's one determinant set to zero; bear with it.

Where it comes from. Think back to last lesson. $A\vec v = \lambda\vec v$ is the same as $(A - \lambda I)\vec v = \vec 0$, and we want a nonzero $\vec v$ there. A square matrix sends a nonzero vector to zero exactly when it isn't invertible, and that's exactly when its determinant is zero. So the eigenvalues are the $\lambda$ that make $\det(A - \lambda I) = 0$. Expanding the determinant gives a polynomial in $\lambda$, which is why an $n\times n$ matrix has at most $n$ eigenvalues.

For $2\times2$ matrices: $\lambda^2 - (\operatorname{tr}A)\lambda + \det A = 0$.

Shortcuts:

• Triangular matrices: the eigenvalues are the diagonal entries. That's because $A - \lambda I$ is triangular too, and a triangular determinant is just the product of the diagonal entries.
• The eigenvalues add up to the trace and multiply to the determinant.

Multiplicity. A repeated root has algebraic multiplicity greater than 1. Its eigenspace may be smaller than that, which matters for diagonalization, as you'll see in the next lesson.

A common mistake. It's tempting to subtract $\lambda$ from every entry of $A$, because "$A$ minus $\lambda$" sounds like it. It isn't: $\lambda I$ has $\lambda$ on the diagonal and $0$ everywhere else, so only the diagonal entries change.

There's an animation further down that slides $\lambda$ along and shows exactly where the determinant hits zero.

Engineers find natural frequencies by solving characteristic equations; control engineers place a system's eigenvalues (poles) to make it stable and responsive.