Eigenvalues and Eigenvectors

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Eigenvalues and Eigenvectors
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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.
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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
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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.
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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.
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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.
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0
What would be the Eigenvalue of the red line?
What about the blue line?
What would happen if the Eigenvalue was 1?
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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.
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Find the eigenvalues and corresponding
eigenvectors of matrix A.
Corresponds to eigenvalue of 5
Corresponds to eigenvalue of -1