SolitaryRoad.com

Website owner:  James Miller


[ Home ] [ Up ] [ Info ] [ Mail ]

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




We will present here results for vectors over complex n-space, Vn(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 Vn(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 ole.gif .




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 ole1.gif i.e. the conjugate of the vector


           ole2.gif   


is


           ole3.gif  .





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



                 ole4.gif   


and


                  ole5.gif  



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


          ole6.gif





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:


ole7.gif




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



                  ole8.gif   


is denoted by ||X|| and defined as



                 ole9.gif ole10.gif






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


              ole11.gif





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


             ole12.gif   





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, ... ,Xm of complex n-space it is orthogonal to the space spanned by them.


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


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


5] A basis of Vnm(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, .... ,Xm 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, .... ,Ym inductively as follows:


             ole13.gif


or, stated more succinctly,


             ole14.gif


             ole15.gif

    


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

ole16.gif  .





             


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


            ole17.gif


where Xi∙Xj is the inner product of Xi and Xj .


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


For a set of complex n-vectors X1, X2, .... ,Xp the determinant of the Gramian matrix |G| has a value |G| ole18.gif 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


                                      ole19.gif  


This is equivalent to the condition



              ole20.gif

 

or


             ole21.gif




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 Vn(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).



More from SolitaryRoad.com:

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Who will go to heaven?

The superior person

On faith and works

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

These things go together

Television

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

On responding to wrongs

Real Christian Faith

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

Sizing up people

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-discipline

Self-control, self-restraint, self-discipline basic to so much in life

We are our habits

What creates moral character?


[ Home ] [ Up ] [ Info ] [ Mail ]