Fast way to compute the determinant of all minors (and minors of minors)

Question

I have an invertible Hermitian matrix and square submatrices of , defined:

I need to know the determinant of every .

Is there a fast way of computing this in MATLAB?

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Here is the bad way to do it:

• Loop over [1:2^K]
• Convert loop index to binary vector vSubset
• Compute det(mtxM(vSubset,vSubset))

This runs slow and seems wasteful, since you can build the determinant of a parent matrix from the determinant of its minors.

Answer 1

One way would be to use the Cholesky factorisation. I use the upper triangular form below, so that

M = U'*U

where ' is adjoint and U is upper triangular. Note that det(M) = square( |det(U)|) and that the determinant of U is the product of its diagonal elements.

We can compute the factor of a matrix obtained from M by appending a row and column like this:

M~ = ( M  n )
     ( n' p )
U~ = ( U  x )
     ( 0  y )

where

U'*x = n 
y = sqrt( p - x'*x)

and so det( M~) = det( M) * ( p - x'*x)

I'm not sure of the best way to use this. There is quite a neat recursive way: in pseudo C code

void det_step( double* U, double det, int high_ix)
{ 
int ix;
  for( ix=high_ix+1; ix<dim; ++ix)
  { // notionally add row, col ix
    // augment U, update det (and store in the output)
    det_step( U, det, ix);
  }
}
void dets( double* M, int dim)
{
int ix;
   for( ix=0; ix<dim; ++ix)
   { // compute U and det for matrix consisting of just row/col ix
     // store det in the output
     det_step( U, det, ix);
   }
}