Find All the Square Roots of a Given 2 by 2 Matrix

Square Roots of a Matrix Problems and Solutions

Problem 513

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
\[A=\begin{bmatrix}
2 & 2\\
2& 2
\end{bmatrix}.\]

 
LoadingAdd to solve later

Proof.

Diagonalize $A$.

We first diagonalize the matrix $A$.
The characteristic polynomial of the matrix $A$ is
\begin{align*}
p(t)=\det(A-tI)=\begin{vmatrix}
2-t & 2\\
2& 2-t
\end{vmatrix}=t(t-4).
\end{align*}
Thus, the eigenvalues of $A$ are $0, 4$.
(Since $A$ has two distinct eigenvalues, it is diagonalizable.)

Let us find eigenvectors.
For the eigenvalue $0$, solving $A\mathbf{x}=\mathbf{0}$, we see that
\[\mathbf{v}=\begin{bmatrix}
1 \\
-1
\end{bmatrix}\] is an eigenvector for $0$.

For the eigenvalue $4$, solving $(A-4I)\mathbf{x}=\mathbf{0}$ yields that
\[\mathbf{u}=\begin{bmatrix}
1 \\
1
\end{bmatrix}\] is an eigenvector for $4$.

Thus the matrix
\[S=\begin{bmatrix}
\mathbf{v} & \mathbf{u}
\end{bmatrix}=\begin{bmatrix}
1 & 1\\
-1& 1
\end{bmatrix}\] diagonalizes the matrix $A$, that is,
\[S^{-1}AS=D,\] where $D$ is the diagonal matrix
\[D=\begin{bmatrix}
0 & 0\\
0& 4
\end{bmatrix}.\]

Determine Square Roots of $A$.

Now suppose that $B$ is a matrix such that $B^2=A$.
We have
\begin{align*}
D=S^{-1}AS=S^{-1}B^2S=(S^{-1}BS)(S^{-1}BS)=(S^{-1}BS)^2=B’^2,
\end{align*}
where we set $B’=S^{-1}BS$.

Observe that
\[B’D=B’B’^2=B’^3=B’^2B’=DB’.\] Since $B’$ commutes with the diagonal matrix $D$, the matrix $B’$ is also diagonal.
(To see this directly, put $B’=\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}$ and compute $B’D$ and $DB’$. Then $B’D=D’B’$ requires $b=c=0$.)

Let $B’=\begin{bmatrix}
a & 0\\
0& d
\end{bmatrix}$.
Since $B’^2=D$, we have
\[\begin{bmatrix}
a^2 & 0\\
0& d^2
\end{bmatrix}=\begin{bmatrix}
0 & 0\\
0& 4
\end{bmatrix},\] hence $a=0$ and $d=\pm 2$.


It follows that a square root of $A$ must be $B=SB’S^{-1}$, where $B’$ is one of
\[\begin{bmatrix}
0 & 0\\
0& 2
\end{bmatrix}, \quad \begin{bmatrix}
0 & 0\\
0& -2
\end{bmatrix}.\]

When $B’=\begin{bmatrix}
0 & 0\\
0& 2
\end{bmatrix}$, we compute
\begin{align*}
B&=SB’S^{-1}=\begin{bmatrix}
1 & 1\\
-1& 1
\end{bmatrix}\begin{bmatrix}
0 & 0\\
0& 2
\end{bmatrix}
\frac{1}{2}\begin{bmatrix}
1 & -1\\
1& 1
\end{bmatrix}\\[6pt] &=\begin{bmatrix}
1 & 1\\
1& 1
\end{bmatrix}.
\end{align*}

Similarly, when $B’=\begin{bmatrix}
0 & 0\\
0& -2
\end{bmatrix}$, we obtain
\[B=\begin{bmatrix}
-1 & -1\\
-1& -1
\end{bmatrix}.\]


In summary, the square roots of the matrix $A$ are
\[\begin{bmatrix}
1 & 1\\
1& 1
\end{bmatrix} \text{ and } \begin{bmatrix}
-1 & -1\\
-1& -1
\end{bmatrix}.\]

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


LoadingAdd to solve later

More from my site

  • A Positive Definite Matrix Has a Unique Positive Definite Square RootA Positive Definite Matrix Has a Unique Positive Definite Square Root Prove that a positive definite matrix has a unique positive definite square root.   In this post, we review several definitions (a square root of a matrix, a positive definite matrix) and solve the above problem. After the proof, several extra problems about square […]
  • No/Infinitely Many Square Roots of 2 by 2 MatricesNo/Infinitely Many Square Roots of 2 by 2 Matrices (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 […]
  • Square Root of an Upper Triangular Matrix. How Many Square Roots Exist?Square Root of an Upper Triangular Matrix. How Many Square Roots Exist? 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 […]
  • Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary MatrixDiagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix Consider the Hermitian matrix \[A=\begin{bmatrix} 1 & i\\ -i& 1 \end{bmatrix}.\] (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, find the eigenvectors. (c) Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix […]
  • Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$ Let \[A=\begin{bmatrix} 1 & 2\\ 4& 3 \end{bmatrix}.\] (a) Find eigenvalues of the matrix $A$. (b) Find eigenvectors for each eigenvalue of $A$. (c) Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that […]
  • Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$ Consider the complex matrix \[A=\begin{bmatrix} \sqrt{2}\cos x & i \sin x & 0 \\ i \sin x &0 &-i \sin x \\ 0 & -i \sin x & -\sqrt{2} \cos x \end{bmatrix},\] where $x$ is a real number between $0$ and $2\pi$. Determine for which values of $x$ the […]
  • Diagonalize a 2 by 2 Symmetric MatrixDiagonalize a 2 by 2 Symmetric Matrix Diagonalize the $2\times 2$ matrix $A=\begin{bmatrix} 2 & -1\\ -1& 2 \end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   Solution. The characteristic polynomial $p(t)$ of the matrix $A$ […]
  • Diagonalize the 3 by 3 Matrix if it is DiagonalizableDiagonalize the 3 by 3 Matrix if it is Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 0 & 1 & 0 \\ -1 &0 &0 \\ 0 & 0 & 2 \end{bmatrix}\] is diagonalizable. If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   How to […]

You may also like...

1 Response

  1. 07/18/2017

    […] $A$. (The less trivial question is that these are the only square roots of $A$. See the post “Find All the Square Roots of a Given 2 by 2 Matrix” […]

Leave a Reply

Your email address will not be published. Required fields are marked *

More in Linear Algebra
Square Roots of a Matrix Problems and Solutions
No/Infinitely Many Square Roots of 2 by 2 Matrices

(a) Prove that the matrix $A=\begin{bmatrix} 0 & 1\\ 0& 0 \end{bmatrix}$ does not have a square root. Namely, show...

Close