[Math] Square root of Diagonal matrix

Question

I cannot find an answer to if it is generally possible to take the square root of a diagonal matrix $A$ by taking the square root of each individual component along the main diagonal, e.g. for a 2-by-2 matrix
$$
\sqrt{A} = \begin{pmatrix}
\sqrt{a_1} & 0 \\
0 & \sqrt{a_2} \\
\end{pmatrix}.
$$
Is this OK to do provided that it is a (square) diagonal matrix?

Answer 1

I assume that you consider matrices with entries in a field $\mathbb{F}$. If square roots $\sqrt{a_i}$ exist in $\mathbb{F}$, then it is ok. However, a diagonal matrix $A$ may have a square root even if the $a_i$ do not square roots in $\mathbb{F}$. An example for $\mathbb{F} = \mathbb{R}$ is $$ A = \begin{pmatrix} -1 & 0 \\ 0 & -1 \\ \end{pmatrix}. $$ In fact, a square root of $A$ is given by $$ B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \\ \end{pmatrix}. $$