Linear Algebra in Hilbert Space

Dual Vectors

Dual Space & Dual Vector

Let  be a Hilbert space. The Hilbert space  is defined as the set of linear maps . We denote elements of  by  where the action of  is:

where  is the inner-product of the vector  with the vector 

The set of maps  is a complex vector space itself, and is called the dual vector space associated with  The vector  is called the dual of 

Orthonormal Basis& Coefficients

Consider a Hilbert space  of dimension  A set of  vectors  is called an orthonormal basis for  if

The Kronecker delta function,  is defined to be equal to 1 whenever  and 0 otherwise.

Every  can be written as

The values of  satisfy  and are called the coefficients of  with respect to basis  .

• Hadamard basis We denote the basis vectors of the Hadamard basis as  and 

Hadamard basis is also an orthonormal basis for .

• The set  is an orthonormal basis for  called the dual basis.

Operators

A linear operator on a vector space  is a linear transforma- tion  of the vector space to itself.

Outer Product & Orthogonal Projector

Outer product is obtained by multiplying  on the right by 

The meaning of such an outer product  is that it is an operator which, when applied to  acts as follows.

Orthogonal Projector: The outer product of a vector  with itself is written  and defines a linear operator that maps

That is, the orthogonal projector  projects a vector  in  to the 1 -dimensional subspace of spanned by 

Let  be an orthonormal basis for a vector space  Then every linear operator  on  can be written as

where 

An operator  is called unitary if 

An operator  in a Hilbert space  is called Hermitean (or self-adjoint ) if 

Spectral Theorem

A normal operator  is a linear operator that satifies

Both unitary and Hermitean operators are normal.

Theorem (Spectral Theorem) For every normal operator  acting on a finite-dimensional Hilbert space  there is an orthonormal basis of  consisting of eigenvectors  of . For every finite-dimensional normal matrix , there is a unitary matrix  such that where  is a diagonal matrix.

Note that  is diagonal in its own eigenbasis:  where  are the eigenvalues corresponding to the eigenvectors  We sometimes refer to  written in its own eigenbasis as the spectral decompositionof  The set of eigenvalues of  is called the spectrum of .

Functions of Operators

Tensor Products

Suppose  and  are Hilbert spaces of dimension  and  respectively. Then the tensor product space  is a new, larger Hilbert space of dimension 

Suppose  is an orthonormal basis for  and is an orthonormal basis for Then

is an orthonormal basis for the space .

Suppose  and  are linear operators on  and  respectively. Then  is the linear operator on  defined by

• Matrix representation

The matrix representation of  is :

Schmidt Decomposition Theorem

Theorem (Schmidt decomposition) If  is a vector in a tensor product space  then there exists an orthonormal basis  for  and an orthonormal basis  for and non-negative real numbers  so that

The coefficients  are called Schmidt coefficients.