Sesquilinear form

In linear algebra , the sesquilinear form (Latin sesqui = one and a half) is a function that assigns a scalar value to two vectors and which is linear in one of its two arguments and semilinear in the other. A classic example is the through

${\displaystyle f((v_{1},\ldots ,v_{n}),(w_{1},\ldots ,w_{n}))={\overline {v}}_{1}w_{1}+\ldots +{\overline {v}}_{n}w_{n}}$

defined mapping , the complex standard scalar product . Here, the dash denotes the complex conjugation . ${\displaystyle f\colon \mathbb {C} ^{n}\times \mathbb {C} ^{n}\to \mathbb {C} }$

The two arguments can come from different vector spaces , but they must be based on a common scalar field ; a sesquilinear form is an illustration ; it is a linear form with respect to one argument and a semi- linear form with respect to the other argument. There are different conventions for the order of linear and semi-linear arguments; in physics it is customary to put the semilinear argument first. ${\displaystyle V,W}$ ${\displaystyle K}$ ${\displaystyle f\colon V\times W\to K}$

Above the real numbers , the concept of the sesquilinear form agrees with that of the bilinear form .

definition

Let there be vector spaces over the complex numbers . ${\displaystyle V,W}$

An illustration

${\displaystyle S\colon V\times W\to \mathbb {C} ,\quad (v,w)\mapsto S(v,w)=\langle v,w\rangle }$

is called sesquilinear form if is semilinear in the first and linear in the second argument, that is ${\displaystyle S}$

• ${\displaystyle \langle v_{1}+v_{2},w\rangle =\langle v_{1},w\rangle +\langle v_{2},w\rangle }$
• ${\displaystyle \langle \lambda v,w\rangle ={\overline {\lambda }}\;\langle v,w\rangle ;}$

and

• ${\displaystyle \langle v,w_{1}+w_{2}\rangle =\langle v,w_{1}\rangle +\langle v,w_{2}\rangle }$
• ${\displaystyle \langle v,\lambda w\rangle =\lambda \,\langle v,w\rangle .}$

There are , and . ${\displaystyle v,v_{1},v_{2}\in V}$ ${\displaystyle w,w_{1},w_{2}\in W}$ ${\displaystyle \lambda \in \mathbb {C} }$

Sometimes linearity in the first and semi-linearity in the second argument is required instead; however, this difference is of a purely formal nature.

This definition can also be generalized to vector spaces over other bodies or modules over a ring as soon as there is an excellent automorphism or at least endomorphism on the basic body or ring

${\displaystyle \lambda \mapsto {\overline {\lambda }}}$

given is. A candidate for such endomorphisms is the Frobenius homomorphism with positive characteristics .

The constant zero mapping is a sesquilinear form, we write . Pointwise sums and scalar multiples of sesquilinear forms are again sesquilinear forms. The set of sesquilinear forms thus forms a vector space. ${\displaystyle S=0}$ ${\displaystyle \mathbb {C} }$

Hermitian sesquilinear form

→ Main article : Hermitian sesquilinear form

A sesquilinear form is called Hermitian, if ${\displaystyle S\colon V\times V\to \mathbb {C} }$

${\displaystyle S(v,w)={\overline {S(w,v)}}}$

applies. This definition is analogous to the definition of the symmetrical bilinear form . The adjective "Hermitesch" is derived from the mathematician Charles Hermite .

Examples

An inner product over a complex vector space is a sesquilinear form with Hermitian symmetry, i.e. even a Hermitian form , see also Krein space .

polarization

statement

The so-called polarization formula plays an important role

${\displaystyle {\begin{aligned}4\cdot S(y,x)&=\sum _{k=0}^{3}\mathrm {i} ^{k}S(x+\mathrm {i} ^{k}y,x+\mathrm {i} ^{k}y)\\&=S(x+y,x+y)+\mathrm {i} S(x+\mathrm {i} y,x+\mathrm {i} y)-S(x-y,x-y)-\mathrm {i} S(x-\mathrm {i} y,x-\mathrm {i} y),\end{aligned}}}$

which shows that the shape is already uniquely determined by its values ​​on the diagonal, ie on pairs of the shape . ${\displaystyle \langle \xi ,\xi \rangle }$

The polarization formula only applies to sesquilinear forms, but not to general bilinear forms.

Special case

A direct consequence of the polarization formula is the fact that the form already disappears when for all . ${\displaystyle S}$ ${\displaystyle S(x,x)=0}$ ${\displaystyle x}$

Or to put it another way: if for everyone , then , well . ${\displaystyle S(x,x)=T(x,x)}$ ${\displaystyle x}$ ${\displaystyle (S-T)(x,x)=0}$ ${\displaystyle S=T}$

Counterexample

This statement does not apply to general bilinear forms, consequently there cannot be a polarization formula. This can be seen in the following example. Be and sit ${\displaystyle V=W\cong \mathbb {R} ^{2}}$

${\displaystyle S(x,y):=x^{T}{\begin{pmatrix}0&-1\\1&0\end{pmatrix}}y=-x_{1}y_{2}+x_{2}y_{1}}$.

${\displaystyle S}$is apparently bilinear and it applies to everyone . On the other hand is . ${\displaystyle S(x,x)=-x_{1}x_{2}+x_{1}x_{2}=0}$ ${\displaystyle x\in \mathbb {R} ^{2}}$ ${\displaystyle S((1,0),(0,1))=1}$

Inference

Let be a Hilbert space and a bounded linear operator . Then is a bounded sesquilinear form. The narrowness means that (here ). Conversely, it follows from Fréchet-Riesz's representation theorem that every bounded sesquilinear form determines a bounded operator such that for all . ${\displaystyle ({\mathcal {H}},\langle \cdot ,\cdot \rangle )}$ ${\displaystyle T}$ ${\displaystyle S(x,y):=\langle Tx,y\rangle }$ ${\displaystyle |S(x,y)|\leq C\|x\|\cdot \|y\|}$ ${\displaystyle C=\|T\|}$ ${\displaystyle T}$ ${\displaystyle S(x,y)=\langle Tx,y\rangle }$ ${\displaystyle x,y\in {\mathcal {H}}}$

In particular, disappears exactly when disappears. This can also be easily seen directly as follows: if so follows for all , so . The converse follows immediately from the definition of . ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle S=0}$ ${\displaystyle \|Tx\|^{2}=S(x,Tx)=0}$ ${\displaystyle x\in {\mathcal {H}}}$ ${\displaystyle T=0}$ ${\displaystyle S}$

With the polarization identity it follows that an operator is zero if and only if for all . However, this statement only applies to the basic field of complex numbers ; the condition that T is self-adjoint is also necessary for the real numbers . ${\displaystyle \langle Tx,x\rangle =0}$ ${\displaystyle x}$ ${\displaystyle \mathbb {C} }$

Sesquilinear forms on modules

The concept of the sesquilinear form can be generalized to any modules , with the complex conjugation being replaced by any anti-automorphism on the underlying, not necessarily commutative ring . Have modules over the same ring and an anti-automorphism . A picture is said to be -Sesquilinearform if for any , and the following conditions apply: ${\displaystyle M,N}$ ${\displaystyle R}$ ${\displaystyle \theta }$ ${\displaystyle R}$ ${\displaystyle \langle \cdot ,\cdot \rangle \colon M\times N\to R}$ ${\displaystyle \theta }$ ${\displaystyle m,m_{1},m_{2}\in M}$ ${\displaystyle n,n_{1},n_{2}\in N}$ ${\displaystyle \lambda \in R}$

• ${\displaystyle \langle m_{1}+m_{2},n\rangle =\langle m_{1},n\rangle +\langle m_{2},n\rangle }$
• ${\displaystyle \langle m,n_{1}+n_{2}\rangle =\langle m,n_{1}\rangle +\langle m,n_{2}\rangle }$
• ${\displaystyle \langle \lambda m,n\rangle =\lambda \langle m,n\rangle }$
• ${\displaystyle \langle m,\lambda n\rangle =\langle m,n\rangle \theta (\lambda )}$

literature

• Siegfried Bosch : Linear Algebra . 3. Edition. Springer textbook, Heidelberg 2006, ISBN 3-540-29884-3 , pp. 245-248 .

Individual evidence

1. ↑ D. Werner: functional analysis 5th, extended edition. Springer, 2004, ISBN 3-540-21381-3 , Korollar V.5.8, p. 236.
2. ^ Nicolas Bourbaki : Algèbre (=  Éléments de mathématique ). Springer , Berlin 2007, ISBN 3-540-35338-0 , chap. 9 , p. 10 .