2. Matrix Mutiplication#
2.1. Interpret by column#
We could consider it as A * a set of column vectors, since each column of C only depends on A * each column of B
For example
2.2. Interpret by row#
Rows of C are combination of rows of B
For example
2.3. Another mutiplication rule#
= Column of * Row ofExample
E.g.
it fits our rule: the columns of
are combinations of columns of , and rows of are combinations of rows of .Since $
$can be decomposed as two
and matrices, we can see that no matter the column space or row space,is a line.
Therefore,
2.4. The Laws for Matrix Operations#
2.4.1. About addition#
2.4.2. About multiplication#
2.4.3. About Powers#
2.5. Block multiplication#
If
Just like regular matrix multiplication.
2.6. Inverse#
For rectangular matrices, the left inverse is not equal to the right inverse, but for a square matrix,
2.6.1. Example#
2.6.2. property#
2.7. Two ways to solve the inverse#
2.7.1. Solve them by the meaning#
2.7.2. Gauss -Jordan (Augmented matrix)#
2.8. Singular matrix (do not have inverse)#
Square matrices do not have an inverse if I can find a vector
2.9. Important Example#
The matrix
Can you find 5 walks of length 3 between nodes 1 and 2?
The real question is why
If there is a 2-step path