Factorizations

• categories: Linear algebra, Definition

We can see the factorization in two ways, as combination of columns or combinations or rows.

1. $A=CR=CU{R}^{\prime }$ . Columns-Rows. In first case we take columns from $A$ in $C$, in second case $C$ are the rows from the A, but we will need another matrix between them
2. $A=LU$. Elimination, solving linear equations for square invertible matrices \pmatrix{2 & 3 \\ 4 & 7} = \pmatrix{1 & 0 \\ 2 & 1}\pmatrix{2 & 3 \\ 0&1}
3. $A=QR$. Gram-Schmidt
4. $S=Q\Lambda Q^T$ - For symmetric matrices eigenvectors are orthogonal and $\Lambda$ is diagonal and real. $(Q\Lambda )Q^T={\sum }_{rank1}=\sum {\lambda }_i{q}_i{q}_i^T$ - every element is simmetric. $S{q}_1={\lambda }_1{q}_1{q}_1^T{q}_1={\lambda }_1{q}_1$
5. $A=X\Lambda X^{-1}$
6. $A=U\Sigma V^{-1}$ . $U$ and $V$ are orthogonal, $\Sigma$ - diagonal