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Let $A$ be an invertible $n \times n$ matrix. Suppose we interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point (this hyperplane does not intersect the origin).

Under this geometric interpretation, $A^{-1}$ has an interesting property: the normal vector to the hyperplane is given by the row sums of $A^{-1}$ (i.e. $A^{-1} \cdot 1$, where $1 = \langle 1, \dots, 1 \rangle^T$).

Within this geometric interpretation of $A$, what other interesting properties does $A^{-1}$ have? Do the individual entries of $A^{-1}$ have geometric meaning? How about the column sums?

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Here is a visual answer for the $2\times 2$ case.

  • Plot the row (or column) vectors, $a_1, a_2$ of $A$ in $\mathcal{R}^2$ to visualize $A$. The area of the parallelogram they form is of course $\det(A)$.
  • In the same space, plot the row (or column) vectors $a^1, a^2$ of $A^{-1}$, and the area of their parallelogram is then $\det(A^{-1}) = 1/ \det(A) $.
  • The relationship between the two illustrates various properties of the matrix inverse.

An example is shown in the picture below, which comes from the matrix (in R notation)

A <- matrix(c(2, 1, 
              1, 2), nrow=2, byrow=TRUE)

In the R package matlib I recently added a vignette illustrating this with the following diagram for this matrix.

inverse diagram

Thus, we can see:

  • The shape of $A^{-1}$ is a $90^o$ rotation of the shape of $A$.

  • $A^{-1}$ is small in the directions where $A$ is large

  • The vector $a^2$ is at right angles to $a_1$ and $a^1$ is at right angles to $a_2$

  • If we multiplied $A$ by a constant $k$ to make its determinant larger (by a factor of $k^2$), the inverse would have to be divided by the same factor to preserve $A A^{-1} = I$.

I wondered whether these properties depend on symmetry of $A$, so here is another example, for the matrix A <- matrix(c(2, 1, 1, 1), nrow=2), where $\det(A)=1$.

enter image description here

It would be interesting to extend this to other properties and to the $3 \times 3$ case, which I leave to others.

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  • $\begingroup$ Not all your observations are true in the general case. The 90° thing for example. You just got lucky with your numbers. For example $A = [[-1, 0], [-3, 2]]$, $A^{-1} = [[-1, -0. ], [-1.5, 0.5]]$ $A_1 \cdot A^{-1}_2 = -1.5 \neq 0 $. Note: internal brackets are rows. $\endgroup$ – user3578468 May 25 at 2:42
  • $\begingroup$ Please provide statements of which of my observations are not true in the general case and why. $\endgroup$ – user101089 May 25 at 22:13
  • $\begingroup$ But I did give you an example. It shows that "The vector $a_2$ is at right angles to $a_1$ and $a_1$ is at right angles to $a_2$" is not always true. $\endgroup$ – user3578468 May 27 at 10:50
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It turns out that an answer for the $3 \times 3$ case has similar properties and is also illuminating.

  • Start with a unit cube, representing the identity matrix. Show its transformation by a matrix $A$ as the corresponding transformation of the cube.

  • This also illustrates the determinant, det(A), as the volume of the transformed cube, and the relationship between $A$ and $A^{-1}$.

In R, using the matlib and rgl package, the unit cube is specified as

library(rgl)
library(matlib)
# cube, with each face colored differently
colors <- rep(2:7, each=4)
c3d <- cube3d()
# make it a unit cube at the origin
c3d <- scale3d(translate3d(c3d, 1, 1, 1),
               .5, .5, .5)

A $3 \times 3$ matrix $A$ with $\det(A)=2$ is

A <- matrix(c( 1, 0, 1, 
               0, 2, 0,  
               1, 0, 2), nrow=3, ncol=3)

Extending the 2D idea from the answer above of drawing the images of $A$ and $A^{-1}$ together to 3D, we get the following, best viewed as an animated graphic. The faces of the parallelpiped representing $A^{1}$ are colored identically to those of $A$, so you can see the mapping from one to the other.

inv-demo.gif

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  • $\begingroup$ Great visualization! $\endgroup$ – Vincent Oct 17 '18 at 13:32

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