Eigenvectors

A is a real symmetric nxn matrix with n orthonormal eigenvectors

\vec{e_i}

& corresponding eigenvalues

\lambda_i

. Obtain coefficients

a_i

such that

\vec{x} = \sum_{i} a_i \vec{e_i}

is a solution to

A\vec{x} - \mu \vec{x} = \vec{f}

where

\vec{f}

is a given vector and

\mu

a scalar NOT equal to an eigenvalue of A.

How on earth do you go about dealing with something as generalised as this. I''ve manged to use what's given to get to

A\vec{x} - \mu \vec{x} = \sum_{i} (\lambda_i - \mu) a_i \vec{e_i}

but I have no clue hwo to proceed to find coefficients

a_i

Reply 1

Glutamic Acid

Write

\mathbf{x} = a_1 \mathbf{e}_1 + ... + a_n \mathbf{e}_n

and dot both sides with

\mathbf{e}_i

to obtain

a_i

using orthonormality.

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Eigenvectors

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