이번에 살펴볼 내용은 rank와 관련하여 행렬을 분해하는 방법인 rank factorization 입니다.
자세한 내용은 https://en.wikipedia.org/wiki/Rank_factorization 을 참조하세요.
For all $A$, there are $B$ and $C$ such that
- $A = BC$
- $A \in \mathbb{R^{m \times n}}$, rank $A$ = $r$
- $B \in \mathbb{R^{m \times r}}$, rank $B$ = $r$
- $C \in \mathbb{R^{r \times n}}$, rank $C$ = $r$
-
Let $\mathbf{b}_i \doteq$ i’th basis of $\mathbb{C}(A)$
$\mathbf{a}_j \doteq$ j’th column vector of $A$ -
$\mathbf{a}_j = \sum_{i=1}^r c_{ij} \mathbf{b}_i$
\(\mathbf{a}_j = \begin{align*} \begin{pmatrix} \\ \mathbf{b}_1 & \cdots & \mathbf{b}_r\\ \\ \end{pmatrix} \begin{pmatrix} c_{11}\\ \vdots\\ c_{r1}\\ \end{pmatrix} \end{align*}\) -
$A = BC$
\(\begin{aligned} A = \begin{pmatrix} \\ \mathbf{a}_1 & \cdots & \mathbf{a}_n\\ \\ \end{pmatrix} = \begin{pmatrix} \\ \mathbf{b}_1 & \cdots & \mathbf{b}_r\\ \\ \end{pmatrix} \begin{pmatrix} c_{11} & \cdots & c_{1n}\\ \vdots & \ddots & \vdots\\ c_{r1} & \cdots & c_{rn}\\ \end{pmatrix} = BC \end{aligned}\)