# What is the square root of matrix

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

• @user1551: hmm, I misspoke; I meant nonnegative, not positive. (By contrast, a symmetric matrix with nonnegative eigenvalues always has a square root.) An explicit example is $\left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right]$. – Qiaochu Yuan Jul 12 '16 at 6:28
• There is no such thing as "the" square root of a matrix. A matrix may have infinitely many square roots, or none. A positive definite (symmetric) matrix has a unique positive definite symmetric square root. – Robert Israel Jul 12 '16 at 6:55

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

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 $\epsilon_i=\pm 1$.

• Donald Sullivan's paper from Mathamatics Magazine "The square roots of $2\times 2$ matrices". – marshal craft Jul 12 '16 at 6:51