# Square Root of an Upper Triangular Matrix. How Many Square Roots Exist?

## Problem 133

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)

## Proof.

We will find all matrices $B$ such that $B^2=A$. Such matrices $B$ are square roots of the matrix $A$.

Note that since $A$ is a diagonal matrix, the eigenvalues of $A$ are diagonal entries $1, 4, 9$. Since $A$ has three distinct eigenvalues, it is diagonalizable.
Solving $(A-\lambda I)\mathbf{x}=\mathbf{0}$ for $\lambda=1,4,9$, we find eigenvectors corresponding to eigenvalues $1, 4, 9$ are respectively

$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} , \quad \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}.$

Thus the invertible matrix
$P=\begin{bmatrix} 1 & 1 & 0 \\ 0 &1 &1 \\ 0 & 0 & 1 \end{bmatrix}$ diagonalizes the matrix $A$, that is, we have
$P^{-1} AP=\begin{bmatrix} 1 & 0 & 0 \\ 0 &4 &0 \\ 0 & 0 & 9 \end{bmatrix}.$

Then if $B^2=A$, then we have $(P^{-1}BP)(P^{-1}B)=P^{-1}AP$.
Let $A’=P^{-1}AP$ and $B’=P^{-1}BP$.

Since we have $B’^2=A’$, we have $B’A’=B’^3=A’B’$.
Since $A’$ is diagonal with distinct diagonal entries, this implies that $B’$ is also a diagonal matrix.

A diagonal matrix $B’$ satisfying $B’^2=A’=\begin{bmatrix} 1 & 0 & 0 \\ 0 &4 &0 \\ 0 & 0 & 9 \end{bmatrix}$ is one of
$\begin{bmatrix} \pm 1 & 0 & 0 \\ 0 &\pm 2 &0 \\ 0 & 0 & \pm 3 \end{bmatrix}.$ Hence $B$ must be one of
$P\begin{bmatrix} \pm 1 & 0 & 0 \\ 0 &\pm 2 &0 \\ 0 & 0 & \pm 3 \end{bmatrix}P^{-1}.$ The inverse matrix of $P$ can be calculated as
$P^{-1}=\begin{bmatrix} 1 & -1 & 1 \\ 0 &1 &-1 \\ 0 & 0 & 1 \end{bmatrix}.$ Therefore, all the square roots of the matrix $A$ are
$\begin{bmatrix} 1 & 1 & 0 \\ 0 &1 &1 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} \pm 1 & 0 & 0 \\ 0 &\pm 2 &0 \\ 0 & 0 & \pm 3 \end{bmatrix}\begin{bmatrix} 1 & -1 & 1 \\ 0 &1 &-1 \\ 0 & 0 & 1 \end{bmatrix}$ and we have $8$ square root matrices.

For example, when the diagonal matrix has all positive entries, then one of the square roots is
$\begin{bmatrix} 1 & 1 & 0 \\ 0 &1 &1 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 &0 \\ 0 & 0 & 3 \end{bmatrix}\begin{bmatrix} 1 & -1 & 1 \\ 0 &1 &-1 \\ 0 & 0 & 1 \end{bmatrix}=\begin{bmatrix} 1 & 1 & -1 \\ 0 &2 &1 \\ 0 & 0 & 3 \end{bmatrix}.$

## Related Question.

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

For a solution of this problem, see the post
A Positive Definite Matrix Has a Unique Positive Definite Square Root

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