Website owner: James Miller
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 .
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
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
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:
or, stated more succinctly,
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
The Gramian Matrix. Let X1, X2, .... ,Xp be a set of complex n-vectors. The Gramian matrix is defined as
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| 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
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.
Ayres. Matrices (Schaum).
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 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
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
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-control, self-restraint, self-discipline basic to so much in life
We are our habits
What creates moral character?