= Sesquilinear form
{wiki}
= Sesquilinear
{synonym}
A <sesquilinear form> on a <complex vector space> $V$ is a map $B:V\times V\to\mathbb C$ which, with the convention used throughout this article, is a <linear map> in its first argument and a <conjugate-linear map> in its second. For <vectors> $u,v,w$ and <complex numbers> $\alpha,\beta$, the defining identities are
$$
B(\alpha u+\beta v,w)=\alpha B(u,w)+\beta B(v,w),\qquad
B(u,\alpha v+\beta w)=\overline\alpha B(u,v)+\overline\beta B(u,w).
$$
The bars denote <complex conjugation>. No symmetry or positivity is required. Some authors choose the opposite convention; every formula here keeps the first argument linear.https://www.cs.cornell.edu/courses/cs6210/2025fa/lec/2025-08-25.html[David Bindel, CS 6210: Matrix Computations, Introduction, sesquilinear forms]{ref}
For a <basis> $e_1,\ldots,e_n$ of a <finite-dimensional vector space>, write $u=\sum_i x_i e_i$, $v=\sum_j y_j e_j$, and $A_{ij}=B(e_i,e_j)$. Expanding by the two identities gives
$$
B(u,v)=\sum_{i,j}x_i A_{ij}\overline{y_j}=x^T A\overline y.
$$
Every complex <matrix> $A$ gives a <sesquilinear form> this way. The superscript $T$ means <transpose>, not <conjugate transpose>. Equivalently, the scalar is $y^*A^T x$, where $y^*=\overline y^{\,T}$. This is the convention used for the <matrix of a Hermitian form> when symmetry is imposed. The coordinate formula follows directly by expanding the <basis> coefficients.https://sites.math.washington.edu/~burke/crs/554/lectures/b-lecture4.pdf[Burke, Linear Analysis, Lecture 4, coordinate expansion for sesquilinear forms]{ref}
For a concrete example on $\mathbb C^2$, let $B(z,w)=z_1\overline{w_2}$. Its <matrix> is $\begin{pmatrix}0&1\\0&0\end{pmatrix}$: $B(e_1,e_2)=1$ but $B(e_2,e_1)=0$, so it is not a <Hermitian form>. Also $B((1,i),(1,i))=-i$, so even real diagonal values cannot be assumed for an arbitrary <sesquilinear form>. By comparison, the <bilinear form> $F(z,w)=z_1w_1+z_2w_2$ satisfies $F(iz,w)=F(z,iw)=iF(z,w)$ and has $F((i,0),(i,0))=-1$; it cannot supply positive squared lengths on this <complex vector space>.
The effect of changing the <complex argument> of a multiplier is particularly revealing. For <real numbers> $\theta,\phi$, <sesquilinear forms> satisfy
$$
B(e^{i\theta}u,e^{i\phi}v)=e^{i(\theta-\phi)}B(u,v),\qquad B(\alpha u,\alpha u)=|\alpha|^2B(u,u).
$$
Thus a common change of <complex argument> cancels from the diagonal, whereas the two arguments rotate the output in opposite directions. Conjugating one argument is what replaces the $\alpha^2$ scaling of a <bilinear form> by the nonnegative $|\alpha|^2$ scaling needed for a <Hermitian inner product>. This mechanism also works in <function spaces>: for complex <continuous functions> on $[0,1]$, $B(f,g)=\int_0^1 f(t)\overline{g(t)}\,dt$ obeys the same identities, letting <integration> measure overlap rather than merely multiply coefficients.
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