Kernel of a Gram product ($\ker (T{T}^{\top })=\ker T^{\top }$)

For a real matrix $T$, a Gram product has the displayed null space. Here Gram product means a product $T{T}^{\top }$ or $T^{\top }T$.

Proof. The inclusion from right to left is immediate. Conversely, if $T{T}^{\top }x=0$, thenso $T^{\top }x=0$. 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.