Scalar Products for Complex Vector Spaces

In generalising from real vector spaces to complex vector spaces, we have to be careful with scalar products.

6.2.1 Definition

Let V be a n-dimensional vector space over the complex numbers. We will denote a scalar product, or inner product, of the ordered pair of vectors u, v ∈ V by

h u , v i ∈ C , (6.4)

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where alternative notations are u · v and h u | v i. A scalar product of a vector space over the complex numbers must have the following properties.

(i) Conjugate symmetry, i.e.

h u , v i = h v , u i^∗, (6.5a)

where a^∗is an alternative notation for a complex conjugate (we shall swap between ¯ and^∗freely).

Implicit in this equation is the assumption that for a complex vector space the ordering of the vectors in the scalar product is important (whereas for Rⁿ this is not important). Further, if we let u = v, then (6.5a) implies that

h v , v i = h v , v i^∗, (6.5b)

i.e. h v , v i is real.

(ii) Linearity in the second argument, i.e. for λ, µ ∈ C

h u , (λv1+ µv2) i = λ h u , v1i + µ h u , v2i . (6.5c) (iii) Non-negativity, i.e. a scalar product of a vector with itself should be positive, i.e.

h v , v i > 0 . (6.5d)

This allows us to write h v , v i = kvk², where the real positive number kvk is the norm of the vector v.

(iv) Non-degeneracy, i.e. the only vector of zero norm should be the zero vector, i.e.

kvk²≡ h v , v i = 0 ⇒ v = 0 . (6.5e)

6.2.2 Properties

Scalar product with 0. We can again show that

h u , 0 i = h 0 , u i = 0 . (6.6)

Anti-linearity in the first argument. Properties (6.5a) and (6.5c) imply so-called ‘anti-linearity’ in the first argument, i.e. for λ, µ ∈ C

h (λu1+ µu₂) , v i = h v , (λu₁+ µu₂) i^∗

= λ^∗h v , u1i^∗+ µ^∗h v , u2i^∗

= λ^∗h u₁, v i + µ^∗h u₂, v i . (6.7) Schwarz’s inequality and the triangle inequality. It is again true that

|h u , v i| 6 kuk kvk , (6.8a)

ku + vk 6 kuk + kvk . (6.8b)

with equality only when u is a scalar multiple of v.

6.2.3 Scalar Products in Terms of Components

Suppose that we have a scalar product defined on a complex vector space with a given basis {ei}, i = 1, . . . , n. We claim that the scalar product is determined for all pairs of vectors by its values for all pairs of basis vectors. To see this first define the complex numbers Gij by

G_ij= h e_i, e_ji (i, j = 1, . . . , n) . (6.9) Then, for any two vectors

v =

n

X

i=1

v_ie_i and w =

n

X

j=1

w_je_j, (6.10)

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We can simplify this expression (which determines the scalar product in terms of the Gij), but first it helps to have a definition.

Hermitian conjugates.

Definition. The Hermitian conjugate or conjugate transpose or adjoint of a matrix A = {a_ij}, where a_ij ∈ C, is defined to be

A^†= (A^T)^∗= (A^∗)^T, (6.12)

where, as before,^Tdenotes a transpose.

Example.

Let w be the column matrix of components,

w = Then in terms of this notation the scalar product (6.11) can be written as

h v , w i = v^†G w , (6.15)

where G is the matrix, or metric, with entries Gij.

Remark. If the {e_i} form an orthonormal basis, i.e. are such that

Gij = h ei, eji = δij, (6.16a)

then (6.11), or equivalently (6.15), reduces to (cf. (3.22)) h v , w i =

n

X

i=1

v_i^∗w_i. (6.16b)

Exercise. Confirm that the scalar product given by (6.16b) satisfies the required properties of a scalar product, namely (6.5a), (6.5c), (6.5d) and (6.5e).

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6.3 Linear Maps

Similarly, we can extend the theory of §4 on linear maps and matrices to vector spaces over complex numbers.

Real linear transformations. We have observed that the standard basis, {ei}, is both a basis for Rⁿ and Cⁿ. Consider a linear map T : Rⁿ → R^m, and let T be the associated matrix with respect to the standard bases of both domain and range; T is a real matrix. Extend T to a map

T : Cb ⁿ→ C^m,

where bT has the same effect as T on real vectors. If e_i→ e⁰_i, then as before (bT)ij = (e⁰_i)j and hence T = T .b

Further real components transform as before, but complex components are now also allowed. Thus if v ∈ Cⁿ, then the components of vwith respect to the standard basistransform to components of v⁰ with respect to the standard basisaccording to

v⁰= Tv .

Maps such as bT are referred to as real linear transformations of Cⁿ→ C^m.

Change of bases. Under a change of bases {ei} to {eei} and {fi} to {efi} the transformation law (4.70) follows through for a linear map N : Cⁿ→ C^m, i.e.

N = Ce ⁻¹NA . (6.17)

Remark. If N is a real linear transformation so that N is real, it is not necessarily true that eN is real, e.g. this will not be the case if we transform from standard bases to bases consisting of complex vectors.

Example. Consider the map R : R²→ R² consisting of a rotation by θl; from (4.23)

x

Since diagonal matrices have desirable properties (e.g. they are straightforward to invert) we might ask whether there is a change of basis under which

eR = A⁻¹RA (6.18)

is a diagonal matrix. One way (but emphatically not the best way) to proceed would be to [partially]

expand out the right-hand side of (6.18) to obtain A⁻¹RA = 1

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(A⁻¹RA)21 = sin θ

det A a²₁₁+ a²₂₁
.

Hence eR is a diagonal matrix if a12 = ±ia22 and a21 = ±ia11. A convenient normalisation (that results in an orthonormal basis) is to choose

A = 1

Thence from (4.61) it follows that

ee₁= 1

where we note from using (6.16b) that

hee₁,ee₁i = 1 , hee₁,ee₂i = 0 , hee₂,ee₂i = 1 , i.e. hee_i,ee_ji = δ_ij. (6.20b) Moreover, from (6.3), (6.18) and (6.19) it follows that, as required, eR is diagonal:

Re = 1

Remark. We know from (4.59b) that real rotation matrices are orthogonal, i.e. RR^T= R^TR = I;

however, it is not true that eReR^T= eR^TeR = I. Instead we note from (6.21) that

eReR^†= eR^†R = I ,e (6.22a)

i.e.

eR^† = eR⁻¹. (6.22b)

Definition. A complex square matrix U is said to be unitary if its Hermitian conjugate is equal to its inverse, i.e. if

U^†= U⁻¹. (6.23)

Unitary matrices are to complex matrices what orthogonal matrices are to real matrices. Similarly, there is an equivalent for complex matrices to symmetric matrices for real matrices.

Definition. A complex square matrix A is said to be Hermitian if it is equal to its own Hermitian conjugate, i.e. if

A^†= A . (6.24)

Example. The metric G is Hermitian since from (6.5a), (6.9) and (6.12)

(G^†)ij= G^∗_ji= h ej, eii^∗= h ei, eji = (G)ij. (6.25) Remark. As mentioned above, diagonal matrices have desirable properties. It is therefore useful to know what classes of matrices can be diagonalized by a change of basis. You will learn more about this in in the second part of this course (and elsewhere). For the time being we state that Hermitian matrices can always be diagonalized, as can all normal matrices, i.e. matrices such that A^†A = A A^† . . . a class that includes skew-symmetric Hermitian matrices (i.e. matrices such that A^†= −A) and unitary matrices, as well as Hermitian matrices.