====== matrix thing ======
https://angeloyeo.github.io/2019/07/17/eigen_vector.html
===Eigen things ===
**square matrix ** A : n x n
$ Ax = \lambda x $
x : eigenvector of A corresponding to eigenvalue $\lambda $

**eigenvalue decomposition**
**symmetric** matrix A: n x n
=> there exists an orthonormal basis for $R^n$ consisting of eigenvectors of A
: by definition, $Aq_i = \lambda_i q_i$
=> $ AQ = Q\Lambda $ => $ A = Q\Lambda Q' $, $\Lambda = diag(\lambda_1,...,\lambda_n) $


=== orthogonal ===
orthogonal vector : <x,y> = 0 
orthonormal : if ||x|| = ||y|| = 1

orthogonal matrix : columns are pairwise orthonormal, Q'Q = QQ' = I
=> 1) preserve inner product : (Qx)'(Qy) = x'Q'Qy = X'Iy = x'y
2) preserve 2-norm : $ ||Qx||_2 = \sqrt{ (Qx)'(Qx) } = \sqrt{x'x} = ||X||_2 $
=> transformation that preserves length, but may rotate or reflect the vector about the origin


=== SVD ===
계산기 : https://atozmath.com/MatrixEv.aspx?q=svd

**every matrix (even non-square) ** A : m x n has SVD
$ A = U \Sigma V' $
U : m x m
V : n x n, orthogonal matrices
$ \Sigma $ : m x n, diagonal matrix with signular values of A

only first r = rank(A) singular values are non-zero.

SVD factors provide **eigendecompositions for A'A and AA'**
$ A'A = (U\Sigma V')' U\Sigma V' = V\Sigma U' U\Sigma V' = V \Sigma ' \Sigma V' $
$ AA' = U\Sigma V' (U\Sigma V')' = U\Sigma V' V\Sigma U' = U \Sigma  \Sigma ' U' $
=> $v_i $ : eigenvectors of A'A, $u_i $ : eigenvectors of AA'
and singular values of A $\sigma_i$ are the square root of the eigenvalues of A'A (or equvalently, of AA')


== 계산 ==
{{:data_analysis:pasted:20200419-235513.png}}


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