Re: A determinant problem
- From: Ken Pledger <ken.pledger@xxxxxxxxxxxxx>
- Date: Fri, 22 Jun 2007 13:08:13 +1200
In article <1182451979.888046.210300@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Randy Poe <poespam-trap@xxxxxxxxx> wrote:
On Jun 21, 2:08 pm, lhi...@xxxxxxxxx wrote:
Hi everyone,
I want to calculate the determinant of an NxN matrix algebraically (as
opposed to numerically). Here's the problem:
Let e be a column vector of N ones. Let alpha be some real number.
Let x be a column vector (entries all positive).
So:
e = (1, 1, 1, ..., 1)'
x = (x_1, x_2, ..., x_N)'
Diag(x) is the diagonal matrix with x along the diagonal.
The problem is to find the determinant of the matrix Q, where
Q = alpha . e . e' + (N+1) . Diag(x) - e . x'
....
.... Try applying this rule twice:
http://en.wikipedia.org/wiki/Matrix_determinant_lemma
- Randy
That's nice; but if you prefer, it can also be done quite quickly
by elementary operations. In det(Q), change all the rows but the first
to
r_2 - r_1, r_3 - r_1, ...., r_N - r_1.
Then clear out most of the first column by
c_1 + (x_1/x_2)c_2, c_1 + (x_1/x_3)c_3, ...., c_1 + (x_1/x_N)c_N.
What's left is upper triangular, so just multiply together the diagonal
entries.
Ken Pledger.
.
- References:
- A determinant problem
- From: lhimon
- Re: A determinant problem
- From: Randy Poe
- A determinant problem
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