Inner Product Space

• Inner product spaces add structure to Vector Space by introducing a notion of “distance”.

Basic Properties of Inner Products §

• Let be a vector space over . An inner product on is a function that assigns to every ordered pair a scalar denoted such that and the following hold

  • . The LHS denotes complex conjugation
  • if .

• The following is immediately true for and .

• A vector space with an inner product is called an inner product space.

• (Friedberg 6.1) Let be an inner product space. Then for and

  • .
  • implies that .

• The following is immediately true for and .

  That is, inner products are conjugate linear in the second component.

• Let be an inner product space. The norm of is defined as

• (Friedberg 6.2) Let be an inner product space. Then we have

  • . Otherwise,
  • Cauchy Schwarz Inequality
  • Triangle Inequality

• The distance between and , denoted is defined as

• (Friedberg e6.1.23) The following are true

  • if .

• (Friedberg e6.1.11) The Parallelogram Law

• (Friedberg e6.1.20) The Polar Identitties. where is over

  • If then
  • If then

• The notion of a norm can be generalized to a matrix. Let . The Euclidean norm of is defined by

  Where .

  • (Friedberg 6.36.1) For any square matrix , is finite. and in fact equals , where is the largest eigenvalue of .
  • (Friedberg Lem.6.36.2) For any square matrix , is an eigenvalue of if and only if is an eigenvalue of .
  • (Friedberg 6.36.2) Let be an invertible matrix. Then
    Where is the smallest eigenvalue of .
  • The condition number of is defined as
  • (Friedberg e6.9.10) . In fact, if and only if is a scalar multiple of a unitary or orthogonal matrix.

Dot Product §

• The standard inner product (also called the dot product) is defined such that

• The Cauchy Schwarz Inequality for dot products:

• The Triangle Inequality for dot products:

• (Friedberg Lem.6.12.1) Let , and then

Topics §

• Orthogonality and Orthonormality
• Matrix Conjugate and Adjoint
• Normal Matrix
• Self-Adjoint Matrix
• Definite Matrix
• Unitary and Orthogonal Operators

Links §

• Friedberg, Insel and Spence