What is the quickest way to find whether the eigenvalues of a $2 \times 2$ matrix are inside the unit circle?

Question

The most direct way is to simply calculate the eigenvalues.

However, is there an easier way?

I know that $\lambda’_1 \lambda_2=D$ and $\lambda_1+ \lambda_2 =T$. Can we use this somehow?

Answer 1

Yes, the criterion is that $2 > 1+D > |T|$.

(Not too difficult to show directly from the quadratic formula $\lambda_{1,2}=\frac12 (T \pm \sqrt{T^2-4D})$.)

Answer 2

In general the Gershgorin Theorem is a good way to locate the eigenvalues within circles on the complex plane.

The center of those circles are the diagonal values and the radii are the sum of absolute values of non-diagonal terms on each row or each column.

For example for $\begin {pmatrix} 3&5\\2&1\end {pmatrix}$ eigenvalues are within circles centered at $3$ and $1$ with radii of $5$ and $2$ respectfully.