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In Linear algebra the eigenvectors of a matrixs are these vectors that don't change their direction after applying this matrix( as a Transformation) to the space. But In machine learning (PCA to be specific) the eigenvectors of a matrix are the directions of the maximum variance of the data points in this matrix. I can't connect the two ideas, how can i tell that the direction that doesn't change is the direction of the maximum variance?.

Can i say that the eigenvectors have 2 properties: they have the same direction after applying the transformation, and they also describe the direction of the maximum variance?

Thanks

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  • $\begingroup$ you didn't search 'PCA eigenvectors variance direction', you'll find many explanations, but the key you'll have to understand is that for a symmetric matrix (as the covariance matrix), the SVD is the same as the diagonalization, and the SVD directly gives the principal directions of the rows of some matrix $\endgroup$ – reuns Mar 15 '16 at 20:47
  • $\begingroup$ the largest singular vector is obtained by $\max_{\|u\|^2 \le 1} \|M u\|^2$ while the principal direction by $\min_{\|u\|^2 \le 1} \sum_i \|M_i - u (u^T M_i) \|^2$ . in fact the two are equivalent and give the same result $\endgroup$ – reuns Mar 15 '16 at 20:54
  • $\begingroup$ A minor correction to the above: for a nonnegative definite symmetric matrix (i.e. a symmetric matrix with nonnegative eigenvalues), the SVD is the same as the diagonalization. Otherwise they are slightly different (the SVD needs the diagonal entries to be nonnegative, so you have to move minus signs around). $\endgroup$ – Ian Mar 15 '16 at 21:07
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PCA seeks the linear combination of your variables that has maximum variance, and the assertion is that the coefficients of that best combination constitute an eigenvector for the covariance matrix of the variables. (We assume the coefficients are normalized so that the sum of squares equals 1.)

You can see this in the two-dimensional case: Say you've observed a data set with two variables $x$ and $y$, and suppose the correlation between $x$ and $y$ is $\rho$. For simplicity we can assume the $x$ and $y$ variables each have variance $1$. Then the covariance matrix of $x,y$ is $$ \Sigma:=\begin{pmatrix}1&\rho\\\rho &1\end{pmatrix} $$ and the variance of the linear combination $ax+by$ is $$ \operatorname{var}(ax+by)=a^2\operatorname{var}x+b^2\operatorname{var}y + 2ab\operatorname{cov}(x,y)=a^2+b^2+2\rho ab.\tag1 $$ Suppose we want to maximize (1) over all $a,b$ subject to the constraint that $a^2+b^2=1$. Using Lagrange multipliers, we form the objective function $$ L(a,b;\lambda):= a^2+b^2+2\rho ab+\lambda(a^2+b^2-1) $$ and take partials with respect to $a, b, \lambda$: $$ {\partial L\over\partial a}=2(a+\rho b -\lambda a)\\ {\partial L\over\partial b}=2(b+\rho a -\lambda b)\\ {\partial L\over\partial \lambda}=a^2+b^2-1 $$ The maximum occurs where these partials are zero. Setting the first two of these to zero and rearranging into matrix form, we get: $$ \begin{pmatrix}1-\lambda &\rho\\\rho&1-\lambda\end{pmatrix} \begin{pmatrix}a\\b\end{pmatrix}= \begin{pmatrix}0\\0\end{pmatrix}, $$ or $$ (\Sigma-\lambda I){\bf v}={\bf 0}\tag2 $$ where we write ${\bf v}$ for the column vector $(a,b)^T$. But (2) is saying that $(a,b)^T$ is an eigenvector for the covariance matrix $\Sigma$ corresponding to eigenvalue $\lambda$.

Note that the covariance matrix has two eigenvalues. The corresponding eigenvectors represent the linear combinations with maximum and minimum variance.

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The co-variance can be thought of as defining an ellipsoid. The eigenvectors of the co-variance matrix correspond to the axes of the ellipsoids, and the eigenvalues are the lengths of the axes.

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The principal components in PCA are the linear-algebra eigenvectors of a particular matrix. It is not a different definition of eigenvalues or eigenvectors.

That the components in PCA, defined as solutions of a sequence of optimization problems, coincide with eigenvectors of the covariance matrix is an application of the theory of Rayleigh quotients. See https://en.wikipedia.org/wiki/Min-max_theorem .

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  • $\begingroup$ Yes i know, but how? $\endgroup$ – Misaki Mar 15 '16 at 20:19
  • $\begingroup$ The connection to maximization is the theory of Rayleigh quotients, used as a way to compute eigenvalues. $\endgroup$ – zyx Mar 15 '16 at 20:25

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