Vectors over complex n-space, Inner products, Orthogonal vectors, Triangle Inequality, Schwarz Inequality, Gram-Schmidt orthogonalization process, Gramian Matrix, Unitary matrix, Unitary transformation

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We will present here results for vectors over complex n-space, V_n(C) . Vector elements and scalars are complex numbers from the field of complex numbers, C. Since field R is a subfield of C it is to be expected that each theorem concerning vectors of V_n(C) will reduce to a theorem about vectors in real n-space when real vectors are considered. We will denote the conjugate of a complex number with an over-bar i.e. the conjugate of z is .

Conjugate of a vector. If X is a vector having complex numbers as elements, the vector obtained from X by replacing each element by its conjugate is called the conjugate of X and is denoted by i.e. the conjugate of the vector

Inner (or dot or scalar) product of two complex n-vectors. Let

and

be two vectors whose elements are complex numbers. Then their inner product is given by

Laws governing inner products of complex n-vectors. Let X, Y and Z be complex n-vectors and c be a complex number. Then the following laws hold:

Orthogonal vectors. Two vectors in n-space are said to be orthogonal if their inner product is zero.

Length of a complex n-vector. The length of a complex n-vector

is denoted by ||X|| and defined as

The Triangle Inequality. For two complex n-vectors X and Y

The Schwarz Inequality. For two complex n-vectors X and Y

Theorems

1] Any set of m mutually orthogonal non-zero vectors of complex n-space is linearly independent and spans an m-dimensional subspace of n-space.

2] If a vector is orthogonal to each of the vectors X₁, X₂, ... ,X_m of complex n-space it is orthogonal to the space spanned by them.

3] Let V_n^h(C) be a h-dimensional subspace of a k-dimensional subspace V_n^k(C) of complex n-space V_n(C) where h < k. Then there exists at least one vector X of V_n^k(C) which is orthogonal to V_n^h(R) .

4] Every m-dimensional vector space contains exactly m mutually orthogonal vectors.

5] A basis of V_n^m(C) which consists of mutually orthogonal vectors is called an orthogonal basis. If the mutually orthogonal vectors are also unit vectors, the basis is called a normal or orthonormal basis.

The Gram-Schmidt orthogonalization process. Suppose X₁, X₂, .... ,X_m constitute a basis of some vector space. The Gram-Schmidt orthogonalization process is a procedure for generating from these m vectors an orthogonal basis for the space. The process involves computing a sequence Y₁, Y₂, .... ,Y_m inductively as follows:

or, stated more succinctly,

The vectors Y₁, Y₂, .... ,Y_m given by the above algorithm are mutually orthogonal but not orthonormal. To obtain an orthonormal sequence replace each Y_i by

The Gramian Matrix. Let X₁, X₂, .... ,X_p be a set of complex n-vectors. The Gramian matrix is defined as

where X_i∙X_j is the inner product of X_i and X_j .

A set of vectors X₁, X₂, .... ,X_p are mutually orthogonal if and only if their Gramian matrix is diagonal.

For a set of complex n-vectors X₁, X₂, .... ,X_p the determinant of the Gramian matrix |G| has a value |G| 0. The set of vectors are linearly dependent if and only if |G| = 0.

Unitary matrix. A matrix which is equal to the inverse of its Hermitian conjugate.

The above definition states that a matrix A is unitary if

This is equivalent to the condition

Theorems.

1] The column vectors (or row vectors) of a unitary matrix are mutually orthogonal unit vectors.

2] The column vectors (or row vectors) of an n-square unitary matrix are an orthonormal basis of V_n(C), and conversely.

3] The inverse and transpose of a unitary matrix are unitary.

4] The product of two or more unitary matrices is unitary.

5] The determinant of a unitary matrix has absolute value 1.

Unitary transformation. The linear transformation Y = AX where A is unitary, is called a unitary transformation.

1] A linear transformation preserves lengths (and hence, inner products) if and only if its matrix is unitary.

2] If Y= AX is a transformation of coordinates from the E-basis to another, the Z-basis, then the Z-basis is orthonormal if and only if A is unitary.

References.

Ayres. Matrices (Schaum).