What is the square root of matrix

Question

If matrix A is positive definite and symmetric. Can I use cholesky Factorization to find the square root of A?

:by cholesky factorization ,A=LDL' where L is a low triangular matrix ,D is diagonal matrix, then square root of A is Ld where d is the matrix which square root all the term in D.

Is it correct?

Is there an algorithm for finding square root of A which is not symmetric?

thanks

Answer 1

Algroithm for finding square root of $2 \times 2$ square matrix

See here http://www.maa.org/sites/default/files/Square_Roots-Sullivan13884.pdf

Important is the Cayley-Hamilton theorem which states that a matrix over a commutative ring satisfies it's own charactristic polynomial.

The primary result is:

$$ A = X^2$$

$$X = \epsilon_2 \frac{A+\epsilon_1 \sqrt{det(A)}I}{\sqrt{tr(A)+2\epsilon_1\sqrt{det(A)}}}$$

where $I$ the identity and $\epsilon_i=\pm 1$.