= Kernel of a Gram product
{title2=$\ker(TT^\top)=\ker T^\top$}
For a real <matrix> $T$, a <Gram product> has the displayed <null space>. Here <Gram product> means a product $TT^\top$ or $T^\top T$.
\b[Proof.] The inclusion from right to left is immediate. Conversely, if $TT^\top x=0$, then
$$
0=x^\top TT^\top x=\|T^\top x\|^2,
$$
so $T^\top x=0$. Similarly $\ker(T^\top T)=\ker T$. Thus these <positive semidefinite matrices> retain the relevant <matrix rank> of $T$. The argument uses the positive <Euclidean norm>; a corresponding statement for an arbitrary indefinite <bilinear form> need not hold.
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