Question

Suppose that A is an nxn matrix with an LU factorization, LU. A) What can be said about the diagonal entries of L? B) Express det(A) in terms of the entries of L and U. C) show that A can be row reduced to U using only row replacement operations.

Answer 1

(a) The product of the diagonal entries of L is the determinant of L - likewise for U for that matter. The reason is that the 'other half' of each contains only zeroes.

(b) The determinant of the product of matrices is the product of the determinants. So\[det(a)=det(L)det(U)\]

(c) This is pretty well by definition of row reduction, as our Gauss method is actually symmetric with regard to the 'direction' of the row operations. You could equally start at the bottom row and proceed upwards, using suitably mutated rules for selection of pivots ( now interpreted as the right-most non zero entries in a row ) etc. In that case your 'row reduced' form would be an L style matix, not a U style. And you can back substitute ( Jordan ) from there, without any loss of generality to achieve the same solution set.

Answer 2

I probably ought make a few extra points here. But ignore this if it confuses, it's a rather higher level view essentially connected to the concept of labelling. With the 'atomic' operation used in row reduction - subtract some multiple of one row from another - then I can get any overall matrix pattern I desire. But it's another question again as to whether such a final pattern is useful.

However what 'saves the day' is that when subtracting a multiple of one row from another, the determinant is unaltered. So what is or is not a soluble system does not change by said manipulations. In any case I can turn an L into a U by row and/or column re-ordering operations - and as such swaps can be constructed using 'subtract a multiple of one row/column from another' atomic operations - then likewise will the determinant is constant throughout. A column re-ordering you see is equivalent to changing the name of a variable - which is fine, provided that you remember that you have done so.

What's often not clearly mentioned/appreciated is that row reduction morphs the column space. But that doesn't matter for the usual purposes as the RHS ( in Ax = b say ) also morphs mutatis-mutandis. So the x that solves Ax = b is the very same x that solves Ux = c.

Answer 3

Err, where U was derived from A by row reduction that is, likewise c from b. :-)