Is it possible to reduce the order of matrix A, if eig(A) = eig(A.') = [1;0.3;0;0]? If yes, how?

Question

I have a matrix A

A = [0.7000    0.7000         0         0;
         0         0    0.4000    0.4000;
    0.3000    0.3000         0         0;
         0         0    0.6000    0.6000;]

if I compute the Eigenvalues of this Matrix and it's transpose I'll get:

Eig = [1.0000;
       0.3000;
       0.0000;
       0.0000]

From what I am remembering I should be able to reduce Matrix A to a 2x2 Matrix by somehow transforming it to 2 Dimensional Space. I was thinking of image(A) since there is a 2D kernel mapping to 0. However I don't quite get what I am looking for. In my opinion the result should look like:

A_reduced = [0.7 0.4; 
             0.3 0.6]; 

Is there a mathematical approach that justifies this result? If there is a way, how would I compute it exactly?

Answer 1

Not sure what you are looking for , you could write $A = \begin{bmatrix}0.7 & 0 \\ 0 & 0.4 \\ 0.3 & 0 \\ 0 & 0.6 \end{bmatrix} \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{bmatrix}$.

Generally the SVD is a first port of call for order reduction.