Square root of a $3\times3$ matrix

Question

Here is $3\times3$ matrix$$\begin{pmatrix} 0& 0& 1\\ 0 & -1 & 0\\ 1& 0 & 0\end{pmatrix}$$ How can I solve this by using Cayley-Hamilton? I know how to use Cayley-Hamilton for a $2$-dimensional matrix. How can it help in finding the square root of a $3\times3$ matrix? for 2 dimensional matrix we can solve this equation A^2−(trA)A+(detA)I=0 we have A and I, we can compute det(A^2) so we have det A, and we can find A.

for 2 dimensional matrix using above equation we can compute square root. for example we have this matrix:

A^2= $$\begin{pmatrix} 4& 2\\ 2& 2\end{pmatrix}$$

det A^2= 4

det A=2




4 2             1 0
2 2  +6 *A+ 2*  0 1 =0

by solving above equation we can find A.

Have you tried writing out the characteristic equation? Give us the progress you've had this far. — Jonas Dahlbæk, Jul 08 '14 at 10:39
Your matrix has a negative eigenvalue, so it will not have a positive square root. Are you sure that is what you want to find? How would you find the square root of a $2\times 2$ matrix from the equation you put in the question? — Jonas Dahlbæk, Jul 08 '14 at 11:11
If the matrix is not positive definite, it does not have a square root over $\Bbb{R}$... — Beni Bogosel, Jul 08 '14 at 11:15
I don't see how your approach helps to get the square $S$ of a matrix $M$, such that $S^2=M$... — draks ..., Jul 08 '14 at 11:15

draks ... · Accepted Answer · 2018-06-25T13:25:17.050

4

Since your matrix $M$ is symmetric you can diagonalize it. You'll get $M=U \cdot D \cdot U^\dagger$, where $D$ contains the eigenvalues and $U$ contains the eigenvectors. Take the square roots of the eigenvalues to get $D_{\sqrt \cdot}$ and transform the resulting diagonal matrix $U^\dagger \cdot D_{\sqrt{.}}\cdot U=S$. It's obvious to see that $S^2=M$.

EDIT

And even that I don't understand your original approach, you can use it when you rewrite your matrix to $$ \pmatrix{0 & 1 & 0\\1&0&0\\0&0&-1}. $$ Now you have one $2\times 2$ block suitable for your approach.

edited Jun 25 '18 at 13:25

answered Jul 08 '14 at 11:19

draks ...

18,449

how can i rewrite matrix to that one? – user123 Jul 13 '14 at 18:00
@saeedeh you conjugate $M$ with a matrix $P$ representing the permutation of indices. $PMP^{-1}$ should be the rewritten matrix... – draks ... Jul 14 '14 at 05:23

score -1 · Answer 2 · answered Aug 07 '15 at 21:26

I solved it by Euler's method of solving for the (later named) unit-length eigenvector (a tilted 3D rotational axis), and the Euler 2D angle theta of rotation about the 3D Euler axis.

I did it computationally.

The square root matrix is: { { 0.5 , 0.7071, 0.5 }, {-0.7071,0,0.7071}, {0.5,-.7071,0.5}}

Square root of a $3\times3$ matrix

2 Answers2