Someone offers the following argument

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Conditions of Validity, Ch Two quadratic forms at once: the generalised eigenvalue problem

Problem

Someone offers the following argument. "The matrices

\mathsf{A} = \begin{pmatrix}1&0\\0&-2\end{pmatrix}, \qquad \mathsf{B} = \begin{pmatrix}0&3\\3&0\end{pmatrix}

are both symmetric. By the same reasoning as the mass inner product route, the generalised eigenvalue problem

\mathsf{B}\mathbf{v} = \lambda\mathsf{A}\mathbf{v}

must therefore have two real eigenvalues

\lambda

, and the corresponding eigenvectors must simultaneously diagonalise both

\mathsf{A}

and

\mathsf{B}

, exactly as the modes of an unequal-mass oscillator simultaneously diagonalise

\mathsf{M}

and

\mathsf{K}

." ((a)) Solve

\det(\mathsf{B}-\lambda\mathsf{A})=0

for

\lambda

and use the result to say whether the claim is correct. ((b)) State precisely which hypothesis used for

\mathsf{M}

in the oscillator problem is not met by

\mathsf{A}

, and explain in one or two sentences why that hypothesis, not mere symmetry, is what the diagonalisation argument actually needs.

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