Let ˜k(x, x

0

) = p

1/2

(x)k(x, x

)p

(x

), and assume p(x) > 0 for all x.

Show that the eigenproblem R

˜k(x, x

)φ˜

i(x)dx = λ˜

iφ˜

i(x

) has the same

eigenvalues as R

k(x, x

)p(x)φi(x)dx = λiφi(x

), and that the eigenfunctions are related by φ˜

i(x) = p

(x)φi(x). Also give the matrix version

of this problem (Hint: introduce a diagonal matrix P to take the rˆole of

p(x)). The significance of this connection is that it can be easier to find

eigenvalues of symmetric matrices than general matrices.

Solution.pdf

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