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Is there any quick argument to show that every non-nilpotent matrix $A\in \mathbb C^{2 \times2}$ has a square root? Just the existence without computing it.

Knowing that $A\in \mathbb C^{2 \times2}$ is non-nilpotent basically tells us that at least one eigenvalue is non-zero. But it's far from being diagonalizable.

My text haven't introduced factorisations based on orthogonality yet. So I'm expecting the proof to be based on Jordan form.

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    $\begingroup$ See square roots of $2\times 2$ matrices. $\endgroup$ – Git Gud Apr 27 '14 at 11:22
  • $\begingroup$ @GitGud: nice one, thanks for the link. Yet I doubt that my text expects students to come up with an explicit formula. That's why I asked about an existence argument. $\endgroup$ – Leo Apr 27 '14 at 11:29
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The result is trivial when $A$ is diagonalisable. So, we only need to consider the case where $A$ is non-diagonalisable and non-nilpotent. Hence $A$ has two equal but nonzero eigenvalues $\lambda$.

By Cayley-Hamilton theorem, $A^2=\operatorname{tr}(A)-\det(A)I=2\lambda A-\lambda^2I$. Therefore $A=\frac1{4\lambda}(A+\lambda I)^2$.

Alternatively, as $A$ is invertible, it has a matrix logarithm. Therefore $A=(e^{(\log A)/2})^2$.

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The case where $A$ is diagonalizable is probably know to you.

Assume $A$ isn't diagonalizable. Then, since it is a $2\times 2$ matrix, it's jordan normal form is $\color{grey}{J=}\begin{bmatrix} \lambda & 1\\ 0 & \lambda\end{bmatrix}$, where $\lambda$ is $A$'s only eigenvalue.

It suffices to find a square root of $J$. Assume it exists and denoted it by $\sqrt J$.

Take the ansatz $\sqrt J=\begin{bmatrix} \sqrt \lambda & a\\ b & \sqrt \lambda \end{bmatrix}$, where $\sqrt \lambda$ denotes a square root of $\lambda$.

Then you want $\sqrt J^2=J$, that is $\begin{bmatrix} \lambda +ab & 2a\sqrt \lambda \\ 2b\sqrt \lambda & ab+\lambda \end{bmatrix}=\begin{bmatrix} \lambda & 1\\ 0 & \lambda\end{bmatrix}$, yielding $a=\dfrac 1{2\sqrt \lambda}\land b=0$.

This gives you four different possibilities for $\sqrt J$.

Edit: In retrospective taking $b\neq 0$ in the ansatz is a bit silly. One might as well try to look for upper triangular square roots from the start.

This answer is relevant.

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  • $\begingroup$ Comment, down voter? $\endgroup$ – Git Gud May 17 '14 at 16:47

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