The fundamental geometric transformations

The square matrix A x transforms the vector x into a vector y by the product y=Ax. Multiplication by the matrix A associates two vectors from the Euclidian space fR and therefore corresponds to a geometric transformation in this space. A is a geometric operator. Non-square matrices would associate vectors from Euclidian spaces with different dimensions. The ordered combination of geometric transformations, such as multiple rotations and projections, can be carried out by multiplying in the right order the vector produced at each stage by the matrix associated with the next transformation. 

Rotation matrices can be defined for an arbitrary number of dimensions. They are particularly useful to examine compositional data in three-dimensional spaces in search for regularities unsuspected in two-dimensional spaces. Commercial software (e.g., Systat ) exists that produces geometric transformations in a convenient way. 

If P is a projector, l—P is a projector onto the orthogonal space since (Pxf(I- P)x=xTPTx - xTPTPx=0 

Given the m x n rectangular matrix A, the m x m projector P=A(ATA) lAT projects each m-vector y onto the column-space of A. Py can be written 

As a special case, projection onto a vector b corresponds to the matrix P, such that 

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