Determine the eigenvalues and eigenfunctions of the following
operator (assume σ(x) ≡ 1): L(u) = u''−2u x ∈ (−1,1) with periodic
boundary conditions u(−1) = u(1), u'(−1) = u'(1). Box your final
answer
Determine the eigenvalues and the corresponding normalized
eigenfunctions of the following Sturm–Liouville problem: y''(x) +
λy(x) = 0, x ∈ [0;L], y(0) = 0, y(L) = 0,
Show that Hermitian operators have real eigenvalues. Show that
eigenvectors of a
Hermitian operator with unique eigenvalues are orthogonal. Use
Dirac notation
for this problem.
Find the eigenvalues and eigenfunctions of the given boundary
value problem. Assume that all eigenvalues are real. (Let
n represent an arbitrary positive number.)
y''+λy=
0,
y(0)= 0,
y'(π)= 0
Q3. Consider the matrix A .
Use R statistical software to determine the eigenvalues and
normalized eigenvectors of A, trace of
A, determinant of A, and inverse
of A. Also determine the eigenvalues and
normalized eigenvectors of
A-1. Your answer
should include your R code (annotated with comments) and a
hand-written or typed summary of the answers from the R output.