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 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
and
are given by the normalized eigenvectors of the
matrix (AAT) -1 . See figure. Unit vectors in the direction of
and
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.
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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)
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
and
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
and
.
The lengths of the semi-axes
and
are given by the quantities
and
respectively.
Let a matrix C be given by
The columns of C consist of the vectors
and
.
Let a matrix D be given by
The columns of D consist of the vectors
and
.
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
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
effects stretching /shrinking in the directions of the
and
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
= 1
or, equivalently,
be transformed by the linear point transformation
X' = AX
The equation of the transformed sphere will be
Let
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
Let
Then
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.