All matrix operations conserving sum of matrix elements

Question

My question is somewhat related to Sum of all elements in a matrix

Consider a matrix A, is there any class of matrices, C, such that:

$$grandsum(CA) = grandsum(A)$$, or

$$ e^TCAe = e^TAe $$

That is all the operations where sum of matrix is conserved. What are such C called?

Background: My friend asked me recently regarding a way to represent all arbitrary volume that a chunk of fluid can take in space. His representation includes a square matrix with matrix elements representing solid angles. Deformation of such fluid will conserve volume, which he is representing as sum of individual elements. so while they can change, their sum cannot.

"volume, which he is representing as a sum of elements", shouldn't the absolute value of the determinant be the right choice? — Peter Melech, Jan 09 '20 at 13:41
In this case yes, I was thinking if more general class of such matrices exist — ipcamit, Jan 09 '20 at 13:48

score 3 · Accepted Answer · answered Jan 09 '20 at 13:48

3

Let $B,C$ be any matrices such that $e = (1,1,\ldots,1)$ is an eigenvector of eigenvalue $1$ for $B$. Then the sum of all elements of your matrix $A$ is $e^TAe = (Be)^TA(Ce) = e^T(B^TAC)e$. And hence the matrix $B^TAC$ has the same sum of elements.

answered Jan 09 '20 at 13:48

Lukas Rollier

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Ca you explain a bit more? I mean Would it not always come out to be I? $$Be = \lambda e$$, there are only one unknown in this equation, wont B= \lambda I? – ipcamit Jan 09 '20 at 14:01
B could be the orthogonal projection on $\text{span} (e)$. This clearly does not equal $I$ – Lukas Rollier Jan 09 '20 at 14:03
Or $B = ((2,-1),(0,1))$ works in $\mathbb{C}^{2 \times 2}$ – Lukas Rollier Jan 09 '20 at 14:05

All matrix operations conserving sum of matrix elements

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