2.5 Bilinear forms and inner products
On the real linear space R the following formula defines a bilinear form which serves simultaneously as an inner product: ifx, y∈R, then
BR(x, y) :=xãy . The corresponding (real) quadratic form is
Q:=BR(x, x) =x2 and it defines the (square of the) Euclidean metric onR.
The setCcan be seen both as a real and as a complex linear space and each of these structures generates its own analogue of what we described above.
WhenCis considered as a real linear space, that is,C=R2, then the bilinear form is given, forz=x+iy,w=u+iv, by
BC,R(z, w) :=xu+yv ,
which is exactly the canonical inner product onR2. The corresponding quadratic form is
QC,R(z) :=BC,R(z, z) =x2+y2 and it defines the (square of the) Euclidean metric onC.
WhenCis considered as a complex linear space, then it has both a bilinear and a sesquilinear form:
BC,1(z, w) :=zãw and
BC,2(z, w) :=zãw ,
the second of them being the canonical complex-valued inner product onC. They generate the quadratic forms
QC,1(z) :=BC,1(z, z) :=z2 and
QC,2(z) :=BC,2(z, z) :=|z|2=x2+y2,
where againQC,2(z) =QC,R(z) is the square of the Euclidean metric onC. The forms BC,1 and QC,1 are employed widely in different areas of mathematics, but BC,1is not called usually an inner product andQC,1does not define any metric in the classical sense.
Let us extend these ideas onto the bicomplex context. Starting with the real structure on BC, we see it as R4 ={(x1, y1, x2, y2) = Z} and thus we endow it with the (real) bilinear form
BBC,R(Z, W) :=x1u1+y1v1+x2u2+y2v2,
which is the canonical inner product onR4. The corresponding quadratic form is QBC,R(Z) :=BBC,R(Z, Z) :=x21+y21+x22+y22,
and it defines the (square of the) Euclidean metric onBC.
WhenBCis considered as aC(i)-complex linear space,BC=C2(i), then it has both aC(i)-bilinear and aC(i)-sesquilinear forms:
BBC;i,1(Z, W) :=z1ãw1+z2ãw2=1
2(ZW†+Z†W) and
BBC;i,2(Z, W) :=z1ãw1+z2ãw2= 1
2(ZW∗+Z†W),
the second of them being the canonicalC(i)-valued inner product onC2(i). They generate the respective quadratic forms
QBC;i,1(Z) :=BBC;i,1(Z, Z) :=z12+z22=ZãZ†=|Z|2i and
QBC;i,2(Z) :=BBC;i,2(Z, Z) :=|z1|2+|z2|2= 1
2(ZZ∗+Z†Z)
=1
2(|Z|2k+|Z†|2k) =1
2(|Z|2k+|Z|2k),
where, again, QBC;i,2(Z) = QBC,R(Z) is the square of the canonical Euclidean metric onC2. The formsBBC;i,1 andQBC;i,1 are also widely known and used (for instance, in the theory of complex Laplacian and its solutions called complex harmonic functions), but BBC;i,1 is not called, usually, an inner product on BC, andQBC;i,1 does not define any metric in the classical sense.
The other complex structure onBC, BC=C2(j), is dealt with in the same way. In this context, it is worth writing the corresponding quadratic forms:
QBC;j,1(Z) :=ζ12+ζ22=ZãZ=|Z|2j and
QBC;j,2(Z) :=|ζ1|2+|ζ2|2=1
2(ZZ∗+ZZ†)
=1
2(|Z|2k+|Z†|2k) =1
2(|Z|2k+|Z|2k),
2.5. Bilinear forms and inner products 39 where we notice that the last one coincides withQBC;i,2(Z), and both are equal to the square of the Euclidean metric.
WhenBCis interpreted asD2, the situation is different. Of course, one sets:
BBC;D,1(Z, W) :=z1w1+z2w2=1
2(ZW∗+Z∗W) and
BBC;D,2(Z, W) :=z1w1+z2w2= 1
2(ZW†+Z∗W) = 1
2(ZW +Z∗W†), imitating the previous situations, but now both forms take values inD, not inC or R. Note also that the first of them is hyperbolic bilinear, and the second one can be called hyperbolic sesquilinear; what is more, setting
QBC;D,1(z) :=BBC;D,1(Z, Z) =z21+z22=ZãZ∗=|Z|2k and
QBC;D,2(z) :=BBC;D,2(Z, Z) =z1z1+z2z2
= 1
2(ZZ†+Z∗Z) = 1
2(ZZ+Z∗Z†)
= 1
2(|Z|2i +|Z∗|2i) =1
2(|Z|2j +|Z∗|2j),
one sees that QBC;D,1 has hyperbolic values and that QBC;D,2 takes real values, but it is not positive definite. Thus, the geometry behind them is much more sophisticated. In the next chapter we will elaborate on this.
Finally,BCis a bicomplex module, i.e., a module over itself, which suggests the introduction of a bicomplex bilinear form
BBC(Z, W) :=ZãW and of three bicomplex sesquilinear-type forms:
BBC,bar(Z, W) :=ZãW , (a bar-sesquilinear form), BBC,†(Z, W) :=ZãW†, (a †-sesquilinear form), BBC,∗(Z, W) :=ZãW∗, (a∗-sesquilinear form).
The corresponding quadratic forms coincide with the three “moduli” previously introduced, which take complex or hyperbolic values, making the geometric aspect even more complicated than the above described case of theD-moduleBC=D2. Of course, this makes both cases even more interesting and intriguing.
Let us consider again theR-valued quadratic formQC,2(z) =x2+y2; since it coincides withBC,2(z, z), thenQC,2enjoys the factorization
QC,2= (x+iy)(x−iy) =zãz .
This identity can be seen as one of the reasons for the necessity of introducing complex numbers: if one wants to factorize QC,2(z) (which is a real-valued and positive definite quadratic form; thus, in particular, the set of its values is R- one-dimensional) into the product of two linear forms which should be real two- dimensional, then the imaginary unitiemerges forcedly and generates the whole setC.
A very similar idea is related with the bicomplex numbers. Consider the C(i)-valued quadratic formQBC,i,1(Z) =z12+z22. We know that it factorizes into
QBC,i,1= (z1+jz2)(z1−jz2) =ZãZ†,
where the set of the values of QBC,i,1 is C(i)-one-dimensional but the factors are already C(i)-two-dimensional. Thus, the C(i)-algebra BC arises from a complex quadratic form in the same way as the real algebraCarises from a real quadratic form. Notice that the requirement for the factors to beC(i)-two-dimensional, not one-dimensional, is crucial since without it one has an obvious factorization
z21+z22= (z1+iz2)(z1−iz2) (2.19) which does not serve our purposes.
Let us show that no other number system, but BC, can play the same role.
Assume that there exists a C(i)-two-dimensional commutative algebra such that the four elements of it, say,a, b, c, d, ensure the identity
z12+z22= (az1+bz2)(cz1+dz2) for allz1andz2 inC(i). Hence, for allz1and z2 it holds that
z12+z22=acz12+bdz22+ (ad+bc)z1z2, which is equivalent to
ac= 1; bd= 1; ad+bc= 0.
Thus, all the coefficients are invertible elements and c−1d+d−1c= 0,
i.e.,
c−1d2
=−1.
Therefore, denoting j:=c−1dwe have thatj2 =−1 andj−1=−j; the factoriza- tion becomes
z12+z22= (z1+jz2)(z1−jz2).
Thus, the complex algebra we are looking for should be generated by 1 and by a new elementj=±i, and we have arrived exactly at BC.