I have a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ with positive entries. Its column sum is $k$ and its row sum is $1$, i.e. $\sum col(\mathbf{A}) = k$ and $\sum row(\mathbf{A}) = 1$.
Now looking at $\mathbf{A}^T\mathbf{A}$, I notice that it has a row (and column) sum of $k$, i.e. $\sum row(\mathbf{A}^T\mathbf{A}) = \sum col(\mathbf{A}^T\mathbf{A}) = k$. Is there a reason for this?
Could this be because $\sum row(\mathbf{A}^T\mathbf{A}) = \sum col(\mathbf{A}^T\mathbf{A}) = \sum col(\mathbf{A})*\sum row(\mathbf{A})$?
Thank you
Best Answer
Let $u$ be the column vector of $n$ $1$'s and $v$ the column vector of $m$ $1$'s. The column vector of row sums of $A$ is $A u$ and the row vector of column sums is $v^T A$. You seem to be saying that $v^T A = k u^T$ and $A u = v$. Then $A^T A u = A^T v = (v^T A)^T = k u^T$ and $u^T A^T A = (A u)^T A = v^T A = k u$, so the row and column sums of $A^T A$ are all $k$.