Let $K$ be a $n \times n$ symmetric real positive definite matrix. Let $u$ and $v$ be non-zero $n\times 1$ vectors taking values of $0$'s and $1$'s. Assume that the values in $u$ and $v$ are sorted so that the first $m_u$ or ($m_v$) entries are zeros and the remaining $n-m_u$ or ($n-m_v$) entries are ones. Is the following statement true? If not, what kind of conditions could I impose so that it is true?

$trace\bigg[ (K+ vv^T)^{-1} (K+uu^T)\bigg]=n$ if and only if $v=u$

Obviously, $(\impliedby)$ is true, but I wonder if the other direction holds under some restrictions.


This is not true. E.g. the trace condition is satisfied when $n=2$ and $$ K=\pmatrix{2&1\\ 1&1},\ u=\pmatrix{1\\ 1},\ v=\pmatrix{0\\ 1}. $$ In general, as $(K+vv^T)^{-1}=K^{-1}-\frac{K^{-1}vv^TK^{-1}}{1+v^TK^{-1}v}$. If you put $P=K^{-1}$, using the tracial property, the trace condition can be rewritten as $$ 0=-\frac{v^TPv}{1+v^TPv} +u^TPu-\frac{(u^TPv)^2}{1+v^TPv} $$ or equivalently, $$ (u^TPu)(1+v^TPv)-v^TPv-(u^TPv)^2=0.\tag{1} $$ When $m_u\le m_v$, you may partition $P$ as $$ \pmatrix{\ast&\ast&\ast\\ \ast&B&Z\\ \ast&Z^T&C} $$ where $B$ is $(m_v-m_u)\times(m_v-m_u)$ and $C$ is $(n-m_v)\times(n-m_v)$. Let the sum of all entries in $B$ be $b$ and define $c$ and $z$ analogously for $C$ and $Z$. Condition $(1)$ can then be rewritten as \begin{align} &(b+c+2z)(1+c)-c-(c+z)^2=0\\ \Leftrightarrow\ &b(c+1)=z(z-2).\tag{2} \end{align} In the counterexample above, we have $P=K^{-1}=\pmatrix{1&-1\\ -1&2}$, so that $b=1,c=2$ and $z=-1$.

When $m_u>m_v$, you can partition $P$ analogously, but this time, $B$ is $(m_u-m_v)\times(m_u-m_v)$ and $C$ is $(n-m_u)\times(n-m_u)$. The trace condition in this case becomes \begin{align} &c(1+b+c+2z)-(b+c+2z)-(c+z)^2=0\\ \Leftrightarrow\ &b(c-1)=z(z+2).\tag{3} \end{align} So, even when $u\ne v$, it can happen that $(2)$ or $(3)$ hold when the values of $b,c,z$ are appropriate.

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