Department of Mathematics
Indian Institute of Science
Bangalore 560 012
GRADUATE SEMINAR
Speaker |
: |
Prof. A.R. Shastri |
Affiliation |
: |
IIT, Mumbai (Graduate Seminar) |
Subject Area |
: |
Mathematics
|
Venue |
: |
Lecture Hall - II (Ground floor), Dept of Mathematics
|
Time |
: |
4.00 pm
|
Date |
: |
Tuesday, 23rd May, 2006
|
Event Title1 |
: |
GAUSS ELIMINATION METHOD AND LAGRANGE-BELTRAMI FORMULA |
Abstract |
Let $Q(x_1,\ldots, x_n)$ be a
symemtric (hermitian), positive definite, bilinear form in $n$-variables. A
classical Lagrange-Beltrami formula obtained as an easy corollary of
spectral theorem says that after a change of basis, $Q$ can be expressed as
a `sum of squares' $$(y_1,\ldots, y_n) = \sum_{i=1}^n \delta_i y_i^2.$$It is
well know that the coefficients $\delta_i$ can be expressed in terms of the
leading principal minors of the original symmetric matrix repesenting the
form $Q.$ In this talk, using one single tool viz. Gauss Elimination
Method,we shall give a simple proof of the above result as well as explicit
formula for the change of basis that should be carried out.Related problems
about indefinite forms etc. will be also discussed. The talk is accessible
to all Math students. |