A question regarding matrix $M_{n\times n } (\mathbb{C})$

Question

The following question was asked in my linear algebra quiz and I was not clear about that option. So, I am asking it here. I will write only that option instead of writing whole question.

Let A$\in M_{n\times n } (\mathbb{C})$. Then is the following option true or not (with reasons):

1. There exists B $\in M_{n\times n } (\mathbb{C})$ such that $B^2 =A$ .

I tried by taking a $2\times2$ matrix with $a_{11}=i, a_{22}=i, a_{21}=0, a_{12} =0 $ but $B$ will exist in that case. So, I am not sure what concept should I use now.

Please tell how to proceed.

Answer 1

$A:=\begin{bmatrix}0 & 1\\ 0& 0\end{bmatrix}$ has no square root. On the contrary assume $B^2=A$ for some $B$. Then, $B^4=A^2=0$, so $B$ is nilpotent but $B\not=0$. Since, characteristic polynomial of a $2\times 2$ nilpotent matrix is $x^2$ we have $B^2=0$ by Cayley-Hamilton theorem, a contradiction as $B^2=A\not=0$.

Answer 2

There is a nice reference here which doesn not prove however that ever every invertible matrix has a logrithm, and to see the existance of matrix logrithms, it suffices to show that + has a logrithm, as cited Under what conditions does a matrix $A$ have a square root?

Here's a nice proof of that fact with differential equation theory :

Let $N$ such that $\inf\limits_{m > 0} \lvert\lvert N^{m} \rvert \rvert^{\frac{1}{m}} < 1$ and let $X(t) := I +tN$ defined for $\lvert t \rvert < 1$.

Since $I +tN$ is invertible by Neumann's series, we have that $(I+tN)^{-1} = \sum\limits_{i=0}^{\infty}(-t)^{j}N^{j}$, and it follows that $\begin{cases}X'(t) = N = N(I+tN)^{-1}X(t) \\ X(0) = I\end{cases}$

So since $N$ and $(I+tN)^{-1}$ commutes, we have

$$X(t) = exp(\int_{0}^{t}N(I+\tau N)^{-1})d\tau$$

So for $t=1$

$$I+N = X(1) = exp(\int_{0}^{1}N(I+\tau N)^{-1}d\tau) = \exp(\int_{0}^{1}\sum\limits_{i=1}^{\infty}(-\tau)^{j-1}N^{j}d\tau)$$

$$ = exp(\sum\limits_{i=1}^{\infty}\int_{0}^{1}(-t)^{j-1}N^{j}d\tau) = exp(\sum\limits_{i=1}^{\infty}(-1)^{j-1}\frac{N^{j}}{j}d\tau) = exp(log(I+N))$$

Where we change the order of integration and series since we uniform convergence in $[0,t] \subseteq [0,1]$