Column and Row View of Matrixs

2017.08.22 tutorial math

Swapping of column and rows is one of the basic uses of matrix, once you get a hang of how the matrixes are multiplied, you will get a better idea of what the different matrixes are doing.

Note this is more of a cheatsheet, and it’s more to supplement other materials. This post will make more sense if you have basic grasp of matrixes operations. This just goes a little further into intuitive view of matrixes.

basic idea

The key idea, or the mantra even, is row times column. This rule will come into play rather frequently. By convention, matrixes functions are defined as such.

column view

Column views is to take the columns of a target matrix and then added them up in a weighted fashion.

In ELI5 terms, it means take a 1 times of column1, add to 2 times of column2, etc

Let’s say we have a target 3x3 matrix A. Then we have the 3x1 column multiplier x.

Step 1 structure of matrixs

$\begin{pmatrix} a&b&c \\ d&e&f \\ g&h&i \\ \end{pmatrix} \begin{pmatrix}x \\ y \\ z\end{pmatrix}$

Step 2 weightage of columns

$x \begin{pmatrix}a \\ d \\ g\end{pmatrix} + y \begin{pmatrix}b \\ e \\ h\end{pmatrix} + z \begin{pmatrix}c \\ f \\ i\end{pmatrix}$

swapping columns

From [x y z] to [y z x].

Step 1 I need one of the second column, y

$\begin{pmatrix} x&y&z \end{pmatrix} \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} y \end{pmatrix}$

Step 2 I need one of the third column, z

$\begin{pmatrix} x&y&z \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} y&z \end{pmatrix}$

Step 3 I need one of the first column, x

$\begin{pmatrix} x&y&z \end{pmatrix} \begin{pmatrix} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{pmatrix} = \begin{pmatrix} y&z&x \end{pmatrix}$

row views

Similar to the column view, the row views is a weighted summation of the different rows. (ie. 1 of row 1 + 2 of row 2, etc)

Assume we have a 3x3 target matrix, and 1x3 weightage matrix. The weightage matrix will have to go at the front (basic matrix multiplication rules.

Step 1 - Structure of matrixs

$\begin{pmatrix} x&y&z \end{pmatrix} \begin{pmatrix} a&b&c \\ d&e&f \\ g&h&i \\ \end{pmatrix}$

Step 2 - weighted addition of rows

$x \begin{pmatrix} a&b&c \end{pmatrix} + y \begin{pmatrix} d&e&f \end{pmatrix} + z \begin{pmatrix} g&h&i \end{pmatrix}$

swapping rows

Swapping if rows from (x y z) to (y z x) (both are 3x1 column vectors).

Step 1 - 1 of the second row

$\begin{pmatrix} 0&1&0 \end{pmatrix} \begin{pmatrix} x\\y\\z \end{pmatrix} = \begin{pmatrix} y \end{pmatrix}$

Step 2 - 1 of the third row

$\begin{pmatrix} 0&1&0 \\ 0&0&1 \\ \end{pmatrix} \begin{pmatrix} x\\y\\z \end{pmatrix} = \begin{pmatrix} y\\z \end{pmatrix}$

Step 3 - 1 of the first row

$\begin{pmatrix} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{pmatrix} \begin{pmatrix} x\\y\\z \end{pmatrix} = \begin{pmatrix} y\\z\\x \end{pmatrix}$