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Suppose $n$ vectors $v_1, v_2,...,v_p,v_{p+1},...,v_n$. Let $G(v_1,...,v_n)$ the Gramian matrix of $v_1,...v_n$.

Prove that : $|G(v_1,...,v_p)|\cdot|G(v_{p+1},...v_n)| \geq |G(v_1,...,v_n)|$.

For $p=1$, it could be proved by its positive definiteness, but I have no idea for $p\ge2$...

The definition of Gramian matrix is here: Gram matrix

Thanks a lot ~

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    [Schur complement](https://en.wikipedia.org/wiki/Schur_complement) and [matrix congruence](https://en.wikipedia.org/wiki/Matrix_congruence) are useful here. – 2017-01-26
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    @user1551 I don't really get it for $p>1$...Suppose that 4 block matrices in $|G(v_1,...,v_n)|$ are $A, B, B^T, C$. By using Schur complement, we got $|G(v_1,...v_n)|=|A|\cdot |C-B^TA^{-1}B|$ and I could only prove the case when $C$ is $1\times1$(by positive definiteness) I don't know if it's the same idea for higher dimensions... – 2017-01-26

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