Transformation Matrix
Let’s say our ’data’ is the
̈ ̨ ̈ ̨ 10
Ourbasevectorsare ̋ ‚(ingreen)and ̋ ‚. 01
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2
Transformation Matrix
original transformed
We can stretch along the y-axis and squish along the x-axis with the matrix ̈ ̨
0.5 0 ̋‚
02
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Transformation Matrix
original transformed
We can stretch along the y-axis and squish along the x-axis with the matrix ̈ ̨
0.5 0 ̋‚
02
̈ ̨ ̈ ̨ ̈ ̨
0.5 0 x 0.5x Toseethis,calculate ̋ ‚ ̈ ̋ ‚“ ̋
‚
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0 2 y 2y
Transformation Matrix
original
We rotate counterclockwise
̈ ̨
0 ́1 ̋‚
10
transformed
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Transformation Matrix
original
We rotate counterclockwise
transformed
̈ ̨
0 ́1 ̋‚
10
̈ ̨ ̈ ̨ ̈ ̨
0 ́1 x ́y Toseethis,calculate ̋ ‚ ̈ ̋ ‚“ ̋ ‚
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10yx
Transformation Matrix
original transformed
We can also perform a so-called shear mapping
̈ ̨
1m
̋ ‚. Here m « 1.
01
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Transformation Matrix
original transformed
We can also perform a so-called shear mapping
̈ ̨
1m
̋ ‚. Here m « 1.
01
̈ ̨ ̈ ̨ ̈ ̨
1m x x`my Toseethis,calculate ̋ ‚ ̈ ̋ ‚“ ̋ ‚
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01yy
Transformation Matrix
original
transformed
Notice what happened to the vector ̋ ‚? Nothing. 0
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̈ ̨
1
Transformation Matrix
original
transformed
̈ ̨
1
Notice what happened to the vector ̋ ‚? Nothing. 0
When a vector doesn’t change its direction after multiplying with a matrix, then it’s
an eigenvector.
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Transformation Matrix
original transformed
In our stretching example from earlier, both vectors were actually eigenvectors. They didn’t change the direction.
However, they change the length.
The value by which is changed the length is called the eigenvalue λ.
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Transformation Matrix
original transformed
The largest eigenvalue is λ1 “ 2 and the second largest here is λ2 “ 0.5. ̈ ̨ ̈ ̨
01
The corresponding eigenvectors are v1 “ ̋ ‚and v2 “ ̋ ‚. 10
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Transformation Matrix
original
What is an eigenvector for the matrix
̈ ̨
1m ̋ ‚?
01
transformed
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Transformation Matrix
original
What is an eigenvector for the matrix
transformed
̈ ̨
1m ̋ ‚?
01
̈ ̨
1
We can see that ̋ ‚must be one. What’s the formula?
0
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Formula
A vector v is an eigenvector of the matrix M if M ̈ v “ λv.
λ is the corresponding eigenvalue.
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Formula
A vector v is an eigenvector of the matrix M if M ̈ v “ λv.
λ is the corresponding eigenvalue. ̈ ̨
22 5 ́1
1 Wecanverifythatv1 “ ̋ ‚andλ1 “4:
1
̈ ̨ ̈ ̨ ̈ ̨ ̈ ̨
22141
M ̈v1 “ ̋ ‚ ̈ ̋ ‚“ ̋ ‚“4 ̋ ‚“λ1v1
5 ́1 1 4 1
̈ ̨
́2
Consider M “ ̋
‚. ̈ ̨
The second eigenvector is v2 “ ̋ ⃝c -Trenn, King’s College London
‚what’s λ2?
5
16