Suppose $A$ is symmetric, positive semidefinite and all its diagonal entries are strictly positive (real coefficients – even integer if it helps). Suppose that the first $r$ rows of $A$ are linearly independent.
Is it true that the $r$-th leading principal submatrix is full rank (and therefore positive definite)?
(The $r$-th leading principal submatrix is the top-left $r\times r$ submatrix of $A$ – it is certainly positive semidefinite and symmetric)
What if all the entries are strictly positive?
I asked this on math.stackexchange last week but I did not receive any suggestion:
https://math.stackexchange.com/questions/2221146/full-rank-submatrices-of-positive-semidefinite-matrix-with-positive-entries
Best Answer
Yes, this is true. To see this note that for $A$ positive semidefinite, $v^T A v = 0$ if and only if $Av = 0$. For the less obvious direction, write $A = B^TB$ for a real matrix $B$. Then $0 = v^TAv = v^TB^TBv = \lVert Bv\rVert^2$, so $Bv = 0$ and therefore $Av = B^TBv = 0$.
Now let $R$ be the principal submatrix of $A$ in question. Let $u$ be such that $Ru = 0$. Let $v$ be $u$ padded with zeros up to the dimension of $A$. Then $0 = u^TRu = v^TAv$, so $Av = 0$. Since the first $R$ columns (equiv. rows) of $A$ are linearly independent, $v$ must be zero, so $u$ must be zero and $R$ is full rank.