Let $M(n,\mathbb R)$ denote set of all $n\times n$ matrices over $\mathbb R$.Which are true:

Question

1.If $A\in M(2,\mathbb R)$ is nilpotent and non-zero ,then there exists a matrix $B\in M(2,\mathbb R)$ such that $B^2=A$

2.If $A\in M(n,\mathbb R)$ is symmetric and positive definite ,then there exists a symmetric matrix $B\in M(n,\mathbb R)$ such that $B^2=A$

3.If $A\in M(n,\mathbb R)$ is symmetric ,then there exists a symmetric matrix $B\in M(n,\mathbb R)$ such that $B^3=A$

I dont know how to approach these?

score 1 · Answer 1 · answered Dec 07 '14 at 04:15

1

Here are some hints to get you started.

If $A$ is nilpotent and $B^2 = A$, what can you say about higher powers of $B$? And which power of $A$ is guaranteed to give you the zero matrix?
Think about diagonalizing $A$. Can you compute a square root of a diagonal matrix?
Think about diagonalizing $A$. Can you compute a cube root of a diagonal matrix?

answered Dec 07 '14 at 04:15

Hans Engler

what is meant by cube root or square root of a matrix?can u please elaborate – Learnmore Dec 07 '14 at 04:55
for the first part $A^2=0$ – Learnmore Dec 07 '14 at 04:56
1

$B$ is a square root of $A$ if $B^2 = A$. – Hans Engler Dec 07 '14 at 14:02
I managed to come up with a counter-example for part (a) but couldn't proceed in the other two parts. Would you kindly post a complete solution? – Brozovic Jan 10 '20 at 21:12

score 0 · Answer 2 · answered Dec 07 '14 at 04:52

0

If A is $\begin{bmatrix}{0}&{1}\\{0}&{0}\end{bmatrix}$ clearly ${A}{A}=0$ and you can prove that is a counterexample.

answered Dec 07 '14 at 04:52

Santiago Montoya

2 Answers2