$$X_{(m,n)}=\begin{bmatrix} | & | & & | \\ x_1 & x_2 & \dots & x_m \\ | & | & & | \end{bmatrix} =U\Sigma V^T =\begin{bmatrix} | & | & & | \\ u_1 & u_2 & \dots & u_n \\ | & | & & | \end{bmatrix} \begin{bmatrix} \sigma_1 & & \\ & \ddots & \\ & & \sigma_m \\ & & \end{bmatrix} \begin{bmatrix} | & | & & | \\ v_1 & v_2 & \dots & v_m \\ | & | & & | \end{bmatrix}^T $$

where $n$ is the number of points in each $x$, and $m$ is the number of $x$ values.

This can be further expanded into

$$ X = \sigma_1 u_1 v_1^T + \sigma_2 u_2 v_2^T + \ldots + \sigma_m u_m v_m^T$$ (assuming $n>>m$).

We can therefore select the first $r$ terms of this expansion to get a reduced-order approximation of X. I.e. $$ X \approx \tilde{U}\tilde{\Sigma}\tilde{V}^T$$.

Correlation between the rows and columns of X.

U and V are the eigenvectors of the correlation matrices,$X X^T$ and $X^T X$, respectively. Because calculating $U$ and $V$ this way is slow, more optimized methods are available. Which is the inner product of $x_i ^T x_j$ where $i$ and $j$ are the $i$th and $j$th column of $X$. If the inner product is large, it means that $x_i$ and $x_j$ are similar. If small, it means that $x_i$ and $x_j$ are different.

This is the statistical interpretation of SVD.

PCA finds elipsoids (depending on dimensionality) that account for the largest, second largest, etc. variance in your data.

Note that X is defined differently as a matrix of horizontally aligned matrix (instead of vertical).

https://www.youtube.com/watch?v=fkf4IBRSeEc

Steps. 1. Compute mean row, X' 2. Subtract mean from X. $B=X-X'$ 3. Calc covariance matrix, $C=B^T B$ 4. Computer eigs of C. Eigenvectors, V, and eigvenvalues, D. 5. Principal components, T=BV. (V also called “loadings”) If $B=U \Sigma V^T$, then $T=U \Sigma$.

Takeaway, PCA can be found by saving the SVD of (X minus its row-mean).

SVD allows us to generalized Ax=b to non-square A matrices. (pseudo-inverse)