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I have a matrix \begin{align} Y = \left( \begin{array}{cc} c & \mathbf{b}^{\top} \newline \mathbf{b} & X \end{array} \right) \end{align} Both $X$ and $Y$ are positive semi-definite matrices sized $n$-by-$n$ and $(n+1)$-by-$(n+1)$ respectively. $\mathbf{b}$ is an $n$ dimensional vector.

Is it true that $$ rank(Y) - rank(X) \in \{0, 1\} $$ ?

In addition, if I fix $X$ and $\mathbf{b}$, but tune the scalar $c$ so as to minimize $c$ while still keeping $Y$ positive semi-definite. Then at the minimal $c$ is it true that $rank(Y) = rank(X)$?

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