Does anyone have a clear and precise definition of a
Semiseparable Matrix?
AFAIK a separable matrix is a matrix that can be written as an outer product of two vectors (that is a column vector multiplied with a row vector). Then there's a definition that I've found using google:
Here an upper triangular matrix S is called semiseparable if all submatrices taken out of the upper triangular part have rank at most one.
Can someone please explain that to me in other words?
When I delete f.i. the last column and row, I get a submatrix, right? (EDIT: see below) Now this matrix is supposed to have only rank (at most) 1? Wouldn't that mean that the matrix only has 1 linearly independent vector?
EDIT: I think I got it:
When you multiply both vectors x and y you get a matrix where all columns are multiples of y (or x) (*). Drop the (strictly) lower part of the matrix and you get an upper triangular matrix. Take an arbitrary submatrix that do NOT contain any elements (zeros) from below the main diagonal. Because of the linear independence (see *), the matrix only has rank 1 or 0, if it's the zero matrix.
