Furthermore, the triangle inequality for complex norms holds kv − wk ≤ kvk + kwk: Norm and inner products in Cn, We'll prove it later. and abstract inner product spaces Math 130 Linear Algebra The inner product hvjwi of two complex vec- D Joyce, Fall 2013 tors. We would like to have a complex inner We've seen how norms and inner products work product that (1) extends the real product, (2) is in Rn. They can also be defined for Cn. There's a connected to the complex norm by the equation 2 wrinkle in the definition of complex inner products. kvk = hv; vi, and (3) has nice algebraic proper- ties such as bilinearity. The norm of a complex vector v. We'll start In order to get property (2), we'll have to intro- with the norm for C which is the one-dimensional duce a wrinkle into the definition. We cannot de- vector space C1, and extend it to higher dimen- fine h(v1; : : : ; vn)j(w1; : : : ; wn)i as v1w1 + ··· vnwn, sions. because then h(v1; : : : ; vn)j(v1; : : : ; vn)i would equal 2 2 2 2 Recall that if z = x + iy is a complex number v1 + ··· vn which doesn't equal jv1j + ··· + jvnj . If with real part x and imaginary part y, the complex we throw in a complex conjugate, however, it will conjugate of z is defined as z = x − iy, and the work. That explains the following definition. absolute value, also called the norm, of z is defined Definition 1. The standard complex inner prod- as n p p uct of two vectors v and w in C is defined by jzj = x2 + y2 = z z: n hvjwi = h(v1; v2; : : : ; vn)j(w1; w2; : : : ; wn)i Now, if v = (v1; v2; : : : ; vn) is a vector in C n where each vi is a complex number, we'll define its X = v w + v w + ··· v w = v w norm kvk as 1 1 2 2 n n k k k=1 kvk = k(v1; v2; : : : ; vn)k It follows that for each v 2 Cn, our desired con- v u n dition (2) above, holds p 2 2 2 uX 2 = jv1j + jv2j + ··· + jvnj = t jvkj : 2 k=1 kvk = hvjvi: Note that if the coordinates of v all happen to be Also, condition (1) holds. If v and w happen to be real numbers, then this definition agrees with the a real vectors, then their complex inner product is norm for real vector spaces. the same as their real inner product. Norms on Cn enjoy many of the same properties Most of the algebraic properties of complex inner that norms on Rn do. For instance, the norm of products are the same as those of real inner prod- any vector is nonnegative, and the only vector with uct. For instance, inner products distribute over norm 0 is the 0 vector. Also, norms are multiplica- addition, tive in the sense that hujv + wi = hujvi + hujwi; kcvk = jcj kvk and over subtraction, when c is a complex number and v is a complex vector. hujv − wi = hujvi − hujwi; 1 and the inner product of any vector and the 0 vec- (d). hvjvi > 0 if v 6= 0. tor is 0 For an inner product space, the norm of a vector v hvj0i = 0: is defined as kvk = phvjvi. However, complex inner products are not com- Note that when F = R, condition (c) simply says mutative. Instead they have the property that the inner product is commutative. hujvi = hvjui: Properties (a) and (b) state that the inner prod- uct is linear in the first argument. Using those and Complex inner products are linear in their first (c), you can show that the inner product is conju- argument. If c is a complex scalar, then gate linear in the second argument. Condition (d) says that the norm kvk of a vector hcujvi = chujvi is always positive except in the one case the that v = 0. From condition (b) you can infer k0k = 0. In for the second argument, we have instead Another property of norms is that kcvk = jcj kvk. Finally, two more properties of inner products are hujcv = chujvi: the Cauchy-Schwarz inequality jhvjwij ≤ kvk kwk, and the triangle inequality kv + wk ≤ kvk + kwk. The complex conjugate of c comes from our defini- We'll prove them later. tion where we use the complex conjugates of coor- dinates of the second vector. Examples. The standard inner products on Rn In summary, complex inner products are not bi- n linear, but they are linear in the first argument and and C are, of course, the primary examples of in- conjugate linear in the second argument. ner product spaces. Our text describes some other inner product spaces besides the standard ones Rn and Cn. One Abstract linear spaces. So far, we've looked at is a real inner product on the vector space of con- the standard real inner product on Rn and the stan- tinuous real-valued functions on [0; 1]. Another is dard complex inner product on Cn. Although we're an inner product on m × n matrices over either R primarily concerned with standard inner products, or C. We'll discuss those briefly in class. There's there are other inner products, and we should con- another example of the vector space of complex- sider the generalization of these standard inner valued functions on the unit circle we won't have products. We'll call a vector space equipped with time for. an inner product an inner product space. We can make the definitions for abstract inner Math 130 Home Page at product spaces for both the real case and the com- http://math.clarku.edu/~djoyce/ma130/ plex case at the same time. In the definition, we'll take the scalar field F to be either R or C. Definition 2. An inner product space over F is a vector space V over F equipped with a function V × V ! F that assigns to vectors v and w in V a scalar denoted hvjwi, called the inner product of v and w, which satisfies the following four conditions: (a). hu + vjwi = hujwi + hujwi, (b). hcvjwi = chvjwi, (c). hvjwi = hwjvi, and 2.