What are the possible $2\times 2$ semisimple matrices?

Question

I have to find all possible values of $a$, $b$, $c$, and $d$ for which the $2 \times 2$ matrix $\begin{pmatrix}a & b \\ c & d \end{pmatrix}$ is semisimple. I know that a matrix $S$ is semisimple if there is a nonsingular matrix $P$ such that $P^{-1}SP=A$ is diagonal. How can I use this to find all the values $a$, $b$, $c$ and $d$?

Answer 1

Since you already know semisimple is equivalent to diagonalizable (it is, in the case of complex matrices), then you can consider the possible Jordan forms for 2x2 matrices. You want to avoid the situation of a Jordan block. That means the two remaining situations are: you have a multiple of the identity matrix, or the characteristic polynomial has two roots. Both of these situations can be expressed in terms of a, b, c, and d.

Answer 2

In the setting that the underlying field is algebraically closed (which we henceforth assume), a matrix is semisimple iff it is diagonalizable.

If the eigenvalues of a matrix are (pairwise) distinct then the matrix is diagonalizable. If the eigenvalues of a $2 \times 2$ matrix coincide (say, they are both $\lambda$) and it is diagonalizable, it has the form $P^{-1} (\lambda I) P = \lambda I$, i.e., it is already a multiple of the identity matrix.

If we denote the matrix by $$A := \pmatrix{a&b\\c&d},$$ its characteristic polynomial is $p_A(t) = t^2 - (\operatorname{tr} A) t + \det A$, so the eigenvalues of $A$ (roots of $p_A$) coincide iff the discriminant $$\Delta = (\operatorname{tr} A)^2 - 4 \det A = (a - d)^2 + 4 b c .$$ vanishes.

So, $A$ is diagonalizable iff

• $(a - d)^2 + 4 b c$, or
• $a = d$ and $b = c = 0$.

Remark Conversely, we can parametrize the nondiagonalizable $2 \times 2$ matrices: Any such matrix is similar to a $2 \times 2$ Jordan block $J_2(\lambda) := \pmatrix{\lambda&1\\\cdot&\lambda}$, i.e., it is equal to $QJ_2(\lambda)Q^{-1}$ for some (invertible) $Q = \pmatrix{p&q\\r&s}$. Conjugating by $\mu Q$ and $Q$ yield the same matrix, so we may as well assume that $\det Q = ps-qr = 1$, yielding $$\pmatrix{\lambda - pr & p^2 \\ -r^2 & \lambda + pr} .$$