How to recognize linear transformations and projections from matrix operator

Question

I have an object $$ C = \begin{pmatrix} 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \end{pmatrix} $$ where the first row contains x-coordinates for the vertices, the second row contains y-coordinates for the vertices, and the third row contains z-coordinates for the vertices.

I have a matrix operator $$ A = \begin{pmatrix} 2/5 & -1/5 & -1/5 \\ -1/5 & -2/5 & 3/5 \\ -1 & 1 & 0 \end{pmatrix} $$ I have already plotted the object C with the transformed object AC, along with the eigenvectors of A see image.

It looks like A projects C along one eigenvector onto a plane formed by the other two eigenvectors, and also does some reflection/rotation (?) see image.

My questions is this:

By looking at $A$, or its eigenvalues and eigenvectors, how can I recognize what $A$ will do to $C$? I.e. that it is in fact a projection operator? What features in $A$ determine where the object gets projected onto, gets rotated to, gets translated to ... etc.?

Answer 1

Feeding your $A$ into Octave yields:

>> A
A =

   0.40000  -0.20000  -0.20000
  -0.20000  -0.40000   0.60000
  -1.00000   1.00000   0.00000

Then determining eigenvectors and values:

>> [B,C] = eig(A)
B =

  -4.0825e-001  5.7735e-001  5.2453e-017
  4.0825e-001  5.7735e-001  -7.0711e-001
  8.1650e-001  5.7735e-001  7.0711e-001

C =

Diagonal Matrix

  1.0000e+000            0            0
            0  -1.9429e-016            0
            0            0  -1.0000e+000

shows (second eigenvector to eigenvalue zero) that for your $A$ we have $$ A \, (1,1,1)^\top = 0 $$

So if you use a coordinate system with one axis along $(1,1,1)^\top$, then $A$ will zero the component in that direction.

$A$ is not a projection, because $A^2 \ne A$.