Math and Architectures of Deep Learning MEAP V02 by unknow

Author:unknow
Language: eng
Format: epub, pdf
Publisher: Manning Publications Co.
Published: 2020-04-28T22:00:00+00:00

(2.25)

We will compute eigenvalues and eigenvectors of A.

Thus,

Here . If necessary, the reader is encouraged to refresh imaginary and complex numbers from high school algebra.

Thus, we have found (as expected) 3 eigenvalues, 1, and . Each of them will yield one eigenvector. Lets compute the eigenvector corresponding to eigenvalue of 1 by way of example.

Thus, is an eigenvector for the eigenvalue 1 for matrix A. So is

for any real k. In fact, if λ, is an eigen value, eigenvector pair for matrix A, then A = λ ⇔ A(k) = λ(k), i.e. λ, (k) is also an eigenvalue, eigenvector pair of A. In other words, one can only determine the eigenvector upto a fixed scale factor. We take the eigenvector to be of unit length (T= 1) without loss of generality.

The eigenvector for our example matrix turns out to be the Z axis. This is not an accident. Our matrix A was, in fact, a rotation about the Z axis. A rotation matrix will always have 1 as an eigenvalue. The corresponding eigenvector will be the axis of rotation. In 3D, the other two eigenvalues will be complex numbers yielding the angle of rotation This is detailed in section 2.14.

Non-zero eigenvectors corresponding to different eigenvalues are linearly independent

Let us prove this to get some insights. Let λ1, 1 and λ2, 2 be eigenvalue, eigenvector pairs for a matrix A with λ1 ≠ λ2. Then

A1 = λ11

A2 = λ22

If possible, let there be two constants α1 and α2 such that

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