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Matrix - Question

PostPosted: Thu Jun 25, 2015 11:34 pm
by Kowshik90
If a Symmetric Matrix A is defined as A=A^t (A transpose ) and A= {(3,5,2);(5,12,7);(2,7,5)} . Then What is the Minimum Eigen Value of A- Matrix ?

Re: Matrix - Question

PostPosted: Tue Nov 03, 2015 10:53 pm
by SubashBoss
The Characteristic polynomial of the matrix A is
(Taking 'R' as roots symbol - lamda)

R^3 + 20*R^2 - 41*R = 0

Thus by taking a R as common,
R( R^2 + 20*R - 41) = 0

Hence, clearly the minimum Eigen value is 0

A shortcut for this kind of problems, if you find the constant term as 0 in the characteristic poly., then definitely
one of the eigen value is 0 and obviously that would be minimum.
Char. Poly
R^3 - R^2(sum of diagonal elements) + R(sum of minors of diagonal elements) - (determinant(A)) = 0
R^3 - R^2(sum of eigen values) + R(sum of eigen value pair product) - (product of eigen values) = 0

Kindly reply in case you need anymore clarification or if I've done anything wrong.

Happy Learning.. :)