Let $X$ be $v\times b$ matrix with $X1_{b}=r1_{v}$ and $X^T1_{v}=k1_{b}$ where $r,k$ are scalars. Then can we say that $I_v-\frac1{kr}XX^T$ is positive semi definite?
I was thinking in the lines of projections. If we prove that $B:=\frac1{kr}XX^T$ is a projection matrix. Then $I-B=B^\perp$ which is psd. However, I can not prove it. So, any help/suggestion are welcome.
First, observe that $\dfrac{1}{kr}XX^T$ is a stochastic (indeed, doubly stochastic) matrix, since $XX^T\mathbf{1}_v = kX\mathbf{1}_b = kr\mathbf{1}_v$.
We know that the largest eigenvalue of a stochastic matrix is $1$, which implies that every eigenvalue of $I_v - \dfrac{1}{kr}XX^T$ is non-negative. This proves that it is positive semidefinite.