How to identify distinct eigenvalues?

Question

I am having difficulty grasping the definition of 'distinct eigenvalues' and how to identify them. By this, I mean what makes these eigenvalues distinct and is there a procedure for identifying them in a diagonal matrix? Any clarification would be greatly appreciated.

Answer 1

Example. If the square matrix $A$ has the characteristic polynomial

$p_A( \lambda)= \lambda( \lambda-1)^3(\lambda -2)^2$, then the eigenvalues of $A$ (counted with algebraic multiplicities) are

$\lambda_1=0, \lambda_2= \lambda_3=\lambda_4=1, \lambda_5= \lambda_6=2.$

The distinct eigenvalues of $A$ are $0,1, 2.$

Answer 2

When eigenvalues are not distinct, it means that an eigenvalue appears more than once as a root of the characteristic polynomial. In geometric terms, it means that there are multiple linearly independent vectors that the matrix scales by the same constant. For instance, the dilation $\mathbb{R}^2 \to \mathbb{R}^2$ where $\langle 1, 0 \rangle \mapsto \langle 2, 0 \rangle$ and $\langle 0, 1 \rangle \mapsto \langle 0, 2 \rangle$ has eigenvectors $\langle 1, 0 \rangle$ and $\langle 0, 1 \rangle$, both of whose eigenvalues are $2$. So this transformation does not have distinct eigenvalues.

As a general rule, if $\alpha$ is a root of a polynomial $f \in \mathbb{R}[x]$, then $\alpha$ has multiplicity greater than $1$ $\iff \alpha$ is also a root of the derivative $f'$ of $f$. To see why this is true, suppose $\alpha$ is a repeated root, allowing us to write $f(x) = (x-\alpha)(x-\alpha)g(x)$ for some $g \in \mathbb{R}[x]$. As a consequence of the product rule, you'll see that $(x-\alpha)$ also divides $f'$, indicating that $\alpha$ is a root of $f'$ as well. This gives a fast, convenient method with which to check whether a given eigenvalue is repeated, though you can also use polynomial long division and check if $\alpha$ is also a root of $\displaystyle \frac{f(x)}{x-\alpha}$.

Note that the "derivative test" will only tell you whether $\alpha$ is a repeated root; you might need polynomial long division to determine its exact multiplicity. I say "might" because context clues can give away information. For instance, suppose your characteristic polynomial is of degree $3$ and you've found two roots, one of which has multiplicity $2$. Then the remaining eigenvalue cannot be repeated as the total number of roots (counting multiplicity) is at most equal to the degree of the characteristic polynomial (and it is equal when you're working over an algebraically closed field such as the complex numbers). Likewise, if your characteristic polynomial is of degree $n$ and you've found $n$ eigenvalues, then none can repeat per this sum-of-multiplicities fact.