Does there exist a matrix square root of a given matrix ?

Question

Let us call $A$ the following matrix $$ A= \frac13\begin{bmatrix}7 & 2 & -4\\2 & 4 & -2\\-4 & -2 & 7 \end{bmatrix}$$

Is it possible to write $B$ as a polynomial of $A$ ?

The first part of the question is based on the fact that $A$ is diagonizable with all positive eigenvalues (4, 1 and 1). I have no clue how to do the second part, I think it could somehow use the spectral decomposition but I'm not sure. Any help would be appreciated.

See also http://math.stackexchange.com/questions/349721/square-root-of-positive-definite-matrix and http://math.stackexchange.com/questions/313564/square-root-of-a-real-matrix. — Dietrich Burde, Feb 10 '14 at 15:28

score 0 · Accepted Answer · answered Feb 10 '14 at 17:17

The general problem is to evaluate any matrix function $f(A)$, where $f$ is analytical and the domain of $f$ contains the spectrum of $A$).

If the matrix $A$ is symmetric (normal) and/or has unique eigenvalues $\lambda_k$,$k=1,..,m$, $f(A)$ can be computed by determining the interpolation polynomial $p(x)$ for the sample value pairs $(\lambda_k,f(\lambda_k))$.

By diagonalization, one easily sees that $p(A)=f(A)$.

Does there exist a matrix square root of a given matrix ?

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