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?


  • 1
    $\begingroup$ @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 $\left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right]$. $\endgroup$ – Qiaochu Yuan Jul 12 '16 at 6:28
  • $\begingroup$ 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. $\endgroup$ – Robert Israel Jul 12 '16 at 6:55

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 $\epsilon_i=\pm 1$.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.