## A Positive Definite Matrix Has a Unique Positive Definite Square Root

## Problem 514

Prove that a positive definite matrix has a unique positive definite square root.

Add to solve laterProve that a positive definite matrix has a unique positive definite square root.

Add to solve later Let $A$ be a square matrix. A matrix $B$ satisfying $B^2=A$ is call a **square root** of $A$.

Find all the square roots of the matrix

\[A=\begin{bmatrix}

2 & 2\\

2& 2

\end{bmatrix}.\]

**(a)** Prove that the matrix $A=\begin{bmatrix}

0 & 1\\

0& 0

\end{bmatrix}$ does not have a square root.

Namely, show that there is no complex matrix $B$ such that $B^2=A$.

**(b)** Prove that the $2\times 2$ identity matrix $I$ has infinitely many distinct square root matrices.

Find **a square root** of the matrix

\[A=\begin{bmatrix}

1 & 3 & -3 \\

0 &4 &5 \\

0 & 0 & 9

\end{bmatrix}.\]

How many square roots does this matrix have?

(*University of California, Berkeley Qualifying Exam*)

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