1. Complex Eigenvalues
kshum
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2. Steps in calculating eigenvalues and eigenvectors
Given a matrix M.
Find the characteristic polynomial.
Find the roots of the characteristic polynomial.
For each eigenvalue  of M, find the non-zero vectors v such that M v =  v.
kshum
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3. Example: flip
A linear transformation L(x,y) given by: L(x,y) = (x, -y)
x  x
y  – y
kshum
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4. Example: shear action
A linear transformation given by L(x,y) = (x+0.25y, y)
x  x y
y  y
kshum
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5. Repeated eigenvalues, one linearly independent eigenvector
What are the eigenvalues of ?
Eigenvectors ?
Solve
\det \begin{bmatrix}1-\lambda& c\\0 & 1-\lambda \end{bmatrix} = 0
( k nonzero )
kshum
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6. Example: Expansion L(x,y) = (ax, ay), for some constant a. x  ax
y  ay
kshum
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7. Repeated eigenvalues, two linearly independent eigenvectors
What are the eigenvalues of ?
Eigenvectors ?
Solve
\det \begin{bmatrix}1-\lambda& c\\0 & 1-\lambda \end{bmatrix} = 0
All non-zero vectors are eigenvector.
kshum
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8. Example: Rotation
Rotation by 90 degrees counter-clockwise: L(x,y) = (– y , x).
x – y
y  x
kshum
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9. Eigenvalues = ? No real root
\det \begin{bmatrix}1-\lambda& c\\0 & 1-\lambda \end{bmatrix} = 0
kshum
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10. Extension to complex vectors and matrix
Given a square matrix A, a non-zero vector v is called an eigenvector of A, if we an find a number , which may be complex, such that
This number  is the eigenvalue of A corresponding to the eigenvector v.
kshum
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11. Complex Eigenvalues
\det \begin{bmatrix}1-\lambda& c\\0 & 1-\lambda \end{bmatrix} = 0
kshum
ENGG2420B