I have two $3 \times 3$ matrices $A$ and $B$. Each matrix is comprised of three orthogonal unit column vectors. I'm trying to gain some intuition for what it means to multiply one of these matrices by the inverse of the other: $$A B^{-1}$$ What does this do geometrically?
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If its columns are orthogonal unit vectors, then a matrix is orthogonal. This means that its inverse is just its transpose. So $AB^{-1}$ is actually $AB^t$. Let's call this matrix $C$.
In fact, the rows of $A$ are also orthogonal unit vectors, so you can think of them as the axes of an orthogonal coordinate system. In the same way, the rows of $B$ also define an orthogonal coordinate system. Let's denote the rows of $A$ by $a_1, a_2, a_3$, and the rows of $B$ by $b_1, b_2, b_3$.
To understand the meaning of $C$, let's take the element $c_{12}$ as an example. By the definition of matrix multiplication, this is the dot product of the first row of $A$ and the second column of $B^t$. But the second column of $B^t$ is the second row of $B$, which is $b_2$. So, in short, $c_{12} = a_1 \cdot b_2$. So $c_{12}$ represents the component of $a_1$ along the axis $b_2$ of the coordinate system defined by $B$.
More generally, $c_{ij}$ is the component of $a_i$ along the axis $b_j$ of the coordinate system defined by $B$.
What's interesting is what happens when you multiply a vector times $AB^{-1}$. The multiplication will convert coordinates from the "$A$" coordinate system to the "$B$" coordinate system. So, this sort of matrix multiplication really just effects a change of coordinates.
