L4: Eigenvectors and Eigenvalues#
As I said before, Eigenvectors are thoes vectors that do not change direction when a linear transformation \(A\) is applied to them. Formally, the eigenvector \(x\) of \(A\) is a vector such that \(Ax = \lambda x\) for some scalar \(\lambda\). If we allied the \(A\) to \(Av\) again, what will happen? The \(x\) will be scaled again, but the direction maintains. Thus, \(A^2x=\lambda^2x\).
Important
If \(A\) is \(n\times n\) matrix with \(n\) indepdent eigenvectors, then any vector \(v\) can be considered as the linear combination of eigenvectors with some coefficients \(c\in\mathbb{R}^n\), i.e., \(v=Xc\). Thus,
Four properties of eigenvectors#
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Image('imgs/l4-m1.png')
For the determinant party, recall the definition of the determinant of a matrix \(A\), I will just show you a beautiful picture from 3blue1brown.
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from IPython.display import Image
Image('imgs/l4-m2.png', width=800)
Four more conlusions about eigenvectors#
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Image('imgs/l4-m3.png', width=800)
The last one tells us an important fact that only the real symmetric matrix has orthogonal eigenvectors. In fact, I am not sure that in what kind of case that a asymmetric matrix \(A\) also satisfies the \(A^TA=AA^T\).
Conclusion3: similar matrices have the same eigenvalues#
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Conclusion4: the diagonal of the triangular matrix is the eigenvalues#
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Image('imgs/l4-m5.png', width=800)
Conclusion5: diagonalization can fasten the computation of power#
Assume a matrix \(A\) has \(n\) independent eigenvectors. Then, we can diagomialize it as \(A=X\Lambda X^{-1}\). Thus,
Conclusion6: markov matrix will be stable after a long time
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Image('imgs/l4-m6.png', width=800)
For the introduction of Multiplicity#
Refer to the Section 1.6 of the book.