Matrix A and
new column, U |
|
Matrix A (n×n) and
new (k.th) column, U (n×1), for
the matrix. • |
Starting from the (known) inverse of A,
calculates the inverse of another, modified matrix, B, in which the
k.th column (only), U, is different. (In the output,
A is called 'A_old', and B is 'A_new'.)
The procedure is:
(a) Let V = A−1U;
(b) Let E identity matrix, except its column k,
which is e(k,k) = 1 / v(k) and,
for i ≠ k,
e(i,k) = -v(i) / v(k);
and (c) Then, B−1 = E A−1.
(InverseUpdate.xlsx)
The advantage is to avoid any direct inversion
(with E a simple, "quasi-identity" matrix).
The procedure is used in, and is typically "linked" to, Linear Programming,
but is general.
If Rand, the matrix elements are generated in
[1, upper] (continuous range).
(Warning: no proviso is included to guarantee
that any of the matrices is non-singular.)
(A related case...) |
