Given that A is an n x n symmetric matrix, and lambda1 and lambda2 are two different eigenvalues of A. Then the eigenspaces E lambda1 and E lambda2 are orthogonal to each other.
True or False?
Given that A is an n x n
yes it is true
Let A be a symmetric n × n matrix and let λ1, λ2 be two distinct eigenvalues of A i.e. λ1 λ2
with associated eigenvectors X, Y respectively. We have seen that λ1 and λ2 must be real since A
is symmetric.
Then AX = λ1X and AY = λ2Y ........................................................(1)
Transposing the first of there results gives
XTAT = λ1XT................................................................................................(2)
(Since for any two matrices the transpose of a product is the product of the transposes in
reverse order.)
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