MITOCW: Matrix Methods in Data Analysis, Signal Processing, and Machine Learning

Textbook: Linear Algebra and Learning from Data (2019), website click here

Problems #

Section I.1 | 18. If $A = CR$, what are the $C$ and $R$ factors of the matrix $\begin{bmatrix} 0 & A \\ 0 & A \end{bmatrix}$?

Notes #

Factoring matrix $A$ #

Factorization $A=CR$, $R=\text{rref}(A)=\text{row-reduced echelon form of A}$

row space of A = $C(A^T)$

$A=\mathbf{CMR}\xrightarrow{\mathbf{C^T}A\mathbf{R^T}}\mathbf{M}=\mathbf{(C^T C)^{-1}(C^T} A\mathbf{R^T)(RR^T)^{-1}}$

Insights on Elimination $A=LU$ $$ A=\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & & & \\ \vdots & & \cdots & \\ a_{m1} & & & \\ \end{bmatrix} =(\text{col}1)(\text{row}1)+ \begin{bmatrix} 0 & 0 & \cdots & 0\\ 0 & & & \\ \vdots & & A_2 & \\ 0 & & & \\ \end{bmatrix}=\cdots $$

$$ LU=\begin{bmatrix} \\ l_1\\ \\ \end{bmatrix} \begin{bmatrix} & u_1^T & \end{bmatrix}+ \begin{bmatrix} \\ l_2\\ \\ \end{bmatrix} \begin{bmatrix} & u_2^T & \end{bmatrix}+\cdots $$

What makes a space? You can do linear combinations here $$ x,y\text{ in a space}\Leftrightarrow cx, x+y\text{ are also in that space} $$

• Column space $C(A)$: dim=r

• Row space $C(A^T )$: dim=r

• Nullspace $N(A)$

• Nullspace $N(A^T )$