Gram product ($T^{\top }T\text{ or }T{T}^{\top }$)

For a complex $m$-by-$n$ matrix $T$, the products $T^{\ast }T$ and $T{T}^{\ast }$ are Gram products; the star denotes the conjugate transpose. For a real matrix, it reduces to the matrix transpose. If the columns of $T$ are $v_1,\dots ,v_n$, then $T^{\ast }T$ is their Gram matrix under the standard complex inner product:Thus a rectangular matrix gives square Hermitian positive semidefinite matrices. The identityproves nonnegativity and gives $\ker (T^{\ast }T)=\ker T$. Similarly, $\ker (T{T}^{\ast })=\ker T^{\ast }$. These are the kernel of a Gram product identities; combined with the rank-nullity theorem, they show that each Gram product has the same matrix rank as $T$.

For example, $T=(12)$ gives $T{T}^{\ast }=5$, while $T^{\ast }T=\left(\begin{array}{cc}1& 2\\ 2& 4\end{array}\right)$. Both have matrix rank one despite their different sizes. The second is singular because the two columns of $T$ are linearly dependent.