Website owner:  James Miller

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

Linear transformation Y=AX viewed as a product of rotations and elongations /contractions in mutually orthogonal directions     


Consider the case of a two dimensional transformation Y = AX and consider what it transforms a unit circle centered at the origin into. It transforms the circle into an eclipse where the semi-major and semi-minor axes of the ellipse correspond to the directions of the mutually perpendicular elongations / contractions. Also, the semi-major and semi-minor axes represent the images of a certain two diameters of the unit circle which also happen to be mutually perpendicular. The linear transformation is equivalent to a rotation of the x-y system to bring the x axis into coincidence with diameter AB, two elongations / compressions in the x and y directions (which stretch and compress the circle into the shape of the ellipse) and a final rotation that carries the ellipse into its final position.

Unit vectors in the direction of ole.gif and ole1.gif are given by the normalized eigenvectors of the matrix (AAT) -1 . See figure. Unit vectors in the direction of ole2.gif and ole3.gif are given by the normalized eigenvectors of the matrix (AT A) -1 . If B is a matrix whose columns consist of the normalized eigenvectors of (AAT) -1 and C is a matrix whose columns consist of the normalized eigenvectors of (AT A) -1 then A can be written as the product:

                                    A = BDC -1

where D is a diagonal matrix. C effects the first rotation, D effects the two perpendicular elongations / compressions, and B effects the final rotation.





Re-examination and re-working of the same problem. The following represents a re-examination and reworking of the same problem done in June 84.

Consider the following point transformation in the plane: Let a unit circle centered at the origin

1)                     ole5.gif

or, in matrix form,


be transformed by the linear point transformation


where A is a non-singular 2x2 matrix.

The equation of the transformed circle will be


or, equivalently,


and will be an ellipse. See figure.

Let λ1 and λ2 be the eigenvalues of matrix (AAT)-1 . Let B be a matrix whose columns consist of the corresponding normalized eigenvectors of the matrix (AAT)-1 . The columns of B will then consist of elementary unit vectors along the axes ole10.gif and ole11.gif .

The lengths of the semi-axes ole12.gif and ole13.gif are given by the quantities ole14.gif and

  ole15.gif   respectively.

Let a matrix C be given by


The columns of C consist of the vectors ole17.gif and ole18.gif .

Let a matrix D be given by


The columns of D consist of the vectors ole20.gif and ole21.gif .

If we rearrange 7) above we arrive at the following factorization of the matrix A:


Since matrix D is orthogonal 8) may be written as



which is our final result.

In the linear transformation


matrices B and DT are both orthogonal matrices effecting rotations of the coordinate system and matrix ole25.gif effects stretching / compressing in two mutually perpendicular directions. Matrix DT takes a point expressed with respect to the x-y coordinate system and expresses it with respect to the B-O-D coordinate system, matrix ole26.gif effects stretching /shrinking in the directions of the ole27.gif and ole28.gif axes, and matrix B takes a point expressed in the B-O-D coordinate system and expresses it in the final x’-y’ coordinate system.      

General case when A is an nxn matrix. We have developed the formula for the factorization of matrix A for the case of two dimensional space. The case for n-dimensional space is a direct extension of the above reasoning.

Let a unit sphere in n-dimensional space

                         ole29.gif  = 1

or, equivalently,


be transformed by the linear point transformation

                                    X' = AX

The equation of the transformed sphere will be



Let λ1, λ2, .... , λn be the n eigenvectors of the matrix (AAT) -1 and B be a matrix whose columns consist of the corresponding normalized eigenvectors of (AAT) -1 .







which is our desired factorization.

Matrices B and DT are both orthogonal matrices effecting “rotations” in n-dimensional space and matrix


effects stretching / compression in n mutually orthogonal directions in n-dimensional space.

More from

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


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?

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