Here’s a topic that might be familiar all the way back to high school mathematics classes. We’re going to use the elementary matrices to manipulate a matrix. Rather than work out abstract formulas, I’m going to take the specific matrix
and perform example manipulations on it to draw conclusions. The effects of elementary matrices are so, well, elementary that it will be clear how they generalize.
So how do we use the matrices to manipulate matrices? Well, we’re using the elementary matrices to change the input or output bases of the linear transformation represented by the matrix. So to change the output basis we’ll multiply on the left by an elementary matrix, while to change the input basis we’ll multiply on the right by the inverse of an elementary matrix — which itself is an elementary matrix of the same kind.
Let’s start with the swaps. On the left we might have
while on the right we might have
On the left, the action of a swap is to swap two rows, while on the right the action is to swap two columns of the matrix.
Next come the scalings. On the left
and on the right
On the left, the action of a scaling is to multiply a row by the scaling factor, while on the right the effect is to multiply a column by the scaling factor.
Finally, the shears. On the left
and on the right
On the left, the shear adds times the th row to the th row, while on the right, the shear adds times the th column to the th column.
So in general we see that acting on the left manipulates the rows of a matrix, while acting on the right manipulates the columns. We call these the “elementary row operations” and “elementary column operations”, respectively. Any manipulation of the form of a matrix we can effect by these operations can be seen as the result of applying a change of basis matrix on the left (output) or right (input) side. And so any two matrices related by these operations can be seen as representing “the same” transformation in two different bases.