Example of $2\times 2$ matrices with common eigenvalues

Question

Give an example of $2\times 2$ matrices $\mathbf{A, B}$ such that $\forall t \in \mathbb{R}$ the matrix $\mathbf{A} + t\mathbf{B}$ has the eigenvalues $\pm\sqrt{t}$.

I presume that there are no solutions, although I would be glad to hear the tips.

Answer 1

Hint: The polynomial $\lambda^2-t$ happens to have $\lambda=\pm \sqrt t$ as roots. Find $A$ and $B$ such that this is the characteristic polynomial of $A+tB$.

Answer 2

First of all, the eigenvalues of $A$ are zero. Suppose $$A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$$therefore$$|\lambda I-A|=\lambda^2-(a+d)\lambda+ad-bc=0$$since the latter equation has only zero roots, we must have $$a=-d\\ad=bc$$Note that if the eigenvalues of $A+tB$ are $\pm\sqrt t$, then those of ${1\over t}(A+tB)={A\over t}+B$ are $\pm {\sqrt{t}\over t}=\pm{1\over \sqrt t}$by tending $t\to \infty$, we conclude that all the eigenvalues of $B$ are also $0$. Therefore both $A$ and $B$ possess a similar form as$$A=\begin{bmatrix}a_{11}&a_{12}\\a_{21}&-a_{11}\end{bmatrix}$$and$$B=\begin{bmatrix}b_{11}&b_{12}\\b_{21}&-b_{11}\end{bmatrix}$$where$$a_{11}^2=-a_{12}a_{21}\\b_{11}^2=-b_{12}b_{21}$$from which we conclude that$$A+tB=\begin{bmatrix}a_{11}+tb_{11}&a_{12}+tb_{12}\\a_{21}+tb_{21}&-a_{11}-tb_{11}\end{bmatrix}$$so the characteristic equation becomes$$|\lambda I-A-tB|=\lambda^2-(a_{11}+tb_{11})^2-(a_{12}+tb_{12})(a_{21}+tb_{21})\\=\lambda^2-t[2a_{11}b_{11}+a_{12}b_{21}+a_{21}b_{12}]$$therefore we obtain the following set of equations $$2a_{11}b_{11}+a_{12}b_{21}+a_{21}b_{12}=1\\a_{11}^2=-a_{12}a_{21}\\b_{11}^2=-b_{12}b_{21}$$which give the most general conditions on $A$ and $B$. As an interesting special case take$$A=\begin{bmatrix}a&a\\-a&-a\end{bmatrix}\\B=\begin{bmatrix}{1\over 4a}&{1\over 4a}\\-{1\over 4a}&-{1\over 4a}\end{bmatrix}$$