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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.

ole4.gif



                                    



                                    











______________________________________________________________________





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,


ole6.gif


be transformed by the linear point transformation


ole7.gif


where A is a non-singular 2x2 matrix.


The equation of the transformed circle will be


ole8.gif


or, equivalently,


ole9.gif


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



ole16.gif


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


Let a matrix D be given by



ole19.gif


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:


ole22.gif



Since matrix D is orthogonal 8) may be written as

 


  ole23.gif



which is our final result.




In the linear transformation



             ole24.gif



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,



             ole30.gif



be transformed by the linear point transformation



                                    X' = AX


The equation of the transformed sphere will be



             ole31.gif

 

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 .



Let


               ole32.gif


Let



             ole33.gif




Then



             ole34.gif  



which is our desired factorization.


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


   ole35.gif  


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



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