Complex number multiplication

The simplest nontrivial example of a complex vector space is C itself. Adding two complex numbers yields a complex number multiplication of a vector by a scalar in this case is just complex multiplication, which yields a complex number (i.e., a vector in C). Mathematicians sometimes call this complex vector space the complex line. One may also consider C as a real vector space and call it the complex plane. See Figure 2.1. 

Postulate A.—3P is linear. By this is meant (i) the vectors of JP are such that we can define the sum of any two of them, the result being also a vector in o> + 6> = c> (ii) they are such that a meaning can be ascribed to multiplication of any vector in by a scalar complex number, the result being also a vector in. In particular,... 

These numbers do not obey all of the laws of the algebra of complex numbers. They add like complex numbers, but their multiplication is not commutative. The general rules of multiplication of n-dimensional hypercomplex numbers were investigated by Grassmann who found a number of laws of multiplication, including Hamilton s rule. These methods still await full implimentation in physical theory. 

In the vector space L defined over the field of real numbers, every operator acting on L does not necessarily have eigenvalues and eigenvectors. Thus for the operation of 7t/2 rotation on a two-dimensional vector space of (real) position vectors, the operator has no eigenvectors since there is no non-zero vector in this space which transforms into a real multiple of itself. However, if L is a vector space over the field of complex numbers, every operator on L has eigenvectors. The number of eigenvalues is equal to the dimension of the space L. The set of eigenvalues of an operator is called its spectrum. 

Lattice complex P (multiplicity 1 multiplicity is the number of equivalent points in the unit cell)... 

To overcome these problems, we must learn another language Chinese. This is what we wilt call the frequency-domain methods. These methods are a little more removed from our mother tongue of English and a little more abstract. But they are extremely powerful and very useful in dealing with realistically complex processes. Basically this is because the manipulation of transfer functions becomes a problem of combining complex numbers numerically (addition, multiplication, etc.). This is easily done on a digital computer. 

Now we shall study the Hilbert scheme of points on the cotangent bundle of a Riemann surface. Let E be a Riemann surface and T E its cotangent bundle. There exists a natural holomorphic symplectic form uc on T E. The multiplication by a complex number on each fiber gives a natural C -action on T E, and with respect to this action we have "(p uJc = tuc for t E C, where we denote the action of t by T E T E. As explained in Theorem 1.10, the Hilbert scheme inherits a holomorphic symplectic form and... 

These steady states are within the physically possible range of T 0 < T < oo) and X(0 < X < 1). This is in contrast to many situations in the physical sciences where equations have multiple roots but only one root is physically acceptable because the other solutions are either outside the bounds of parameters (such as negative concentrations or temperatures) or occur as imaginary or complex numbers and can therefore be ignored. 

Can you determine all the possible sets of three complex numbers that satisfy the same multiplication table ... 

Complex numbers can be added, subtracted, multiplied and divided like real numbers. For example, the multiplication of z by z gives ... 

Quaternions are similar to complex numbers but of the form a + hi + cj.+ dk with one real and three imaginary parts. The addition of these 4-D numbers is fairly easy, but the multiplication is more complicated. How could such numbers have practical application It turns out that quaternions can be used to describe the orbits of pairs of pendulums and to specify rotations in computer graphics. 

Exercise 1.5 (Geometry of multiplication in C) 77zc complex plane can be considered as a two-dimensional real vector space, with basis 1, i. Show that multiplication by any complex number c is a linear transformation. Find the matrix for multiplication by i in the given basis. Find the matrix for multiplication by e, where d is a real number. Find the matrix for multiplication by a + ib. where a and b are real numbers. 

Does addition of functions survive the equivalence Does scalar multiplication (by complex numbers) survive the equivalence Does multiplication of two functions survive the equivalence ... 

In this chapter we introduce complex linear algebra, that is, linear algebra where complex numbers are the scalars for scalar multiplication. This may feel like review, even to readers whose experience is limited to real linear algebra. Indeed, most of the theorems of linear algebra remain true if we replace R by C because the axioms for a real vector space involve only addition and multiplication of real numbers, the definition and basic theorems can be easily adapted to any set of scalars where addition and multiplication are defined and reasonably well behaved, and the complex numbers certainly fit the bill. However, the examples are different. Furthermore, there are theorems (such as Proposition 2.11) in complex linear algebra whose analogues over the reals are false. We will recount but not belabor old theorems, concentrating on new ideas and examples. The reader may find proofs in any number of... 

For example, the real line R is not a complex vector space under the usual multipUcation of real numbers by complex numbers. It is possible for the product of a complex number and a real number to be outside the set of real numbers for instance, (z)(3) = 3i R. So the real line R is not closed under complex scalar multiplication. 

The trivial complex vector space has one element, the zero vector 0. Addition is defined by 0 -I- 0 = 0 for any complex number c, define the scalar multiple of 0 by c to be 0. Then all the criteria of Definition 2.1 are trivially true. For example, to check distributivity, note that for any c e C we have... 

Exercise 2.1 Consider the set of homogeneous polynomials in two variables with real coefficients. There is a natural addition of polynomials and a natural scalar multiplication of a polynomial by a complex number. Show that the set of homogeneous polynomials with these two operations is not a complex vector space. 

This fact will be at the heart of the proof of our main result in Section 6.5. Proof. First, we show that V satisfies the hypotheses of the Stone-Weierstrass theorem. We know that V is a complex vector space under the usual addition and scalar multiplication of functions adding two polynomials or multiplying a polynomial by a constant yields a polynomial. The product of two polynomials is a polynomial. To see that V is closed under complex conjugation, note that for any x e [—1, 1] and any constant complex numbers flo, , a sN[Pg.102]

Intuitively, two groups that are isomorphic are essentially the same, although they may arise in different contexts and consist of different types of mathematical objects. For example, the unit circle in the complex plane is isomorphic as a group to the set of 2 x 2 rotation matrices. See Figure 4.1. One is a set of complex numbers, and one is a set of matrices with real entries, but if we strip away tJieir contexts and consider only how the multiplication operation works, they have identical mathematical structure. 

This character is the same as the character of the representation on by matrix multiplication in fact, these two representations are isomorphic, as the reader may show in Exercise 4.36. This is an example of the general phenomenon that will help us to classify representations finite-dimensional representations are isomorphic if and only if their characters are equal. See Proposition 6.12. Note that while a representation is a relatively complicated object, a character is simply a function from a group to the complex numbers it is remarkable that so much information about the complicated object is encapsulated in the simpler object. 

Now suppose that the representations Vi and V2 are indeed isomorphic. Let T and f denote isomorphisms (of representations) from V to 72- It suffices to show that T must be a scalar multiple of T. Consider the linear transformation T o V2 V2- By Exercise 4.19, this linear transformation is an isomorphism of representations. Hence by Proposition 6.3, there must be a complex number A such that T o 7 = kl, and hence t = XT. Note that because 7 is an isomorphism, A 0. ... 

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