(a) Let be a self-adjoint operator on a complex inner-product space, . Prove that…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Conditions of Validity, Ch The spectral theorem, degeneracy and the matrices that will not diagonalise

Problem

(a) Let

T

be a self-adjoint operator on a complex inner-product space,

T^{\dagger}=T

. Prove that every eigenvalue of

T

is real, and that eigenvectors belonging to distinct eigenvalues are orthogonal. (b) A mass on a critically damped spring obeys

\ddot x + 2\omega_0\dot x + \omega_0^2 x = 0

with

\omega_0 = 4\,\mathrm{rad\,s^{-1}}

. Write this as

\dot{\mathbf{x}} = \mathsf{A}\mathbf{x}

with

\mathbf{x}=(x,\dot x)

. Show that

\mathsf{A}

has a single (repeated) eigenvalue and only one independent eigenvector, so

\mathsf{A}

is not self-adjoint and not diagonalisable. Set

\mathsf{N} = \mathsf{A}+\omega_0\mathsf{I}

, show

\mathsf{N}^2 = \mathsf{0}

, and use the series for

e^{t\mathsf{A}}

to find

x(t)

for release from

x(0)=2\,\mathrm{cm}

at rest. Evaluate

x(0.2\,\mathrm{s})

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