# Find the basis of a symmetric matrix, analytical expression

I have a problem and I am not able, so far, to find the solution.

I am trying to understand if there is a way, given a symmetric matrix $B$ (which is computed as $B=A^TA$ where I know the matrix $A$) to compute one of its basis in a closed form (analytical way).

I can easily solve my problem by the matlab command:

basis = orth(B)

Which gives me a basis of the matrix $B$.

The problem is that matlab, of course, uses a numerical way to find this basis while I was wondering if there is a closed form to express this basis.

orht(B)

• The column vectors of the matrix $B$ span the image (of $Bx$) - i.e. they are the basis of the image after you get rid of the linearly dependent vectors. The kernel is the solution to the homogeneous system of equations, i.e. $Bx=0$. If this doesn't make sense, please edit your question and give a more detailed description of what you need or at least some example. – Michael Medvinsky Oct 29 '15 at 14:04