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Eigenvalues and Eigenvectors

Imagine this

What if I took a position vector (not zero

vector) and multiplied it by a matrix and ended

up with what I started with or a scalar multiple

of what I started with (ie the direction of the

vector is the same or exactly opposite).

Investigate this using Autograph.

Eigenvalues and Eigenvectors

So what we are trying to find is this..

M is a matrix and ? is a scalar constant

Rearranging

In order to factorise we must turn our scalar

into a matrix by multiplying it by the identity

matrix.

We can now factorise

Eigenvalues and Eigenvectors

Write M as a matrix

We can write (M- ?I) in the following way and

then simplify the equation we found earlier.

If the determinant of (M- ?I) was non-zero we

could inverse it and multiply it by the RHS.

Eigenvalues and Eigenvectors

If the determinant of (M- ?I) was non-zero we

could inverse it and multiply it by the RHS.

This would mean that our vector was zero which

contradicts what we want.

This means that the determinant of (M- ?I) must

be zero so is singular.

Eigenvalues and Eigenvectors

This bit is the useful bit to remember

This is called the characteristic equations and

will allow us to find the eigenvalues

(characteristic values)

The Eigenvalues represent the scale factor of the

distance our position vector (which we still need

to find) moves in relation to its original

position. There may be 1, 2 or no real

eigenvalues in a 2x2 matrix.

0

What would be the Eigenvalue of the red line?

What about the blue line?

What would happen if the Eigenvalue was 1?

Eigenvalues and Eigenvectors

Once we have found the Eigenvalues, we can

substitute these back to find their corresponding

Eigenvectors.

This represents the Eigenvector. It is not unique

as any multiple of it would still be an

Eigenvector!

Eigenvectors represent Invariant Lines. These are

the lines of points that map onto themselves

after a transformation.

Find the eigenvalues and corresponding

eigenvectors of matrix A.

Corresponds to eigenvalue of 5

Corresponds to eigenvalue of -1

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