Linear Transformation – Cascaded Connection

Lecture 8:Cascaded Linear TransformationsRow and Column SelectionPermutation MatricesMatrix TransposeSections 2.2.3, 2.3
Linear Transformation – Cascaded Connection
Viewed as a linear transformation, the productABrepresents the cascade (series) connection ofBfollowed by A, as shown below. This means that for a n-dimensional input vector x, the output y is given byy =(AB)x=A(Bx)To see why this is so, takex=e(j),i.e, any standard unit vector inRn, then(AB)e(j)is thejthcolumn of the matrixAB. BydefinitionofAB, theithentry in that column equals the dot product of theithrow of A and thejthcolumn of B.Be(j)is thejthcolumn of B. Thus the (i, j)thentry ofA(Be(j)) equals the dot product of theithrow of A and thejthcolumn of B.Therefore (AB)x=A(Bx) forx=e(j).
AssociativityandCommuntativity
(AB)C=A(BC)=ABCand this extends to products involving four or more matrices.In general, ABBAi.e., matrix multiplication is not commutative—even in cases where both products are well-definedand have the same dimensions (this happens if and only if both A and B are square matrices of the same dimensions). There are, however, exceptions.
IfAis a m × n matrix ande(j)is thejthstandard unit vector in Rn×1, thenAe(j)=jthcolumn ofAThis follows directly from thedefinitionof the matrix-vector product, whereAxis a linear combination of the columns ofAwithcoefficientsgiven by the respective entries ofx.
Example
Supposethen,
Permutation
The columns of anm × nmatrix A are permuted by selecting all of them in a particular order. This amounts to taking the productAP, wherePis a permutation matrix, i.e.,the columns ofPare the standard n-dimensional unit vectors (in any order); equivalently,the rows ofPare the standard n-dimensional unit vectors (in any order).Example,
Symmetry
IfAism × n, its transposeATis the n × m matrix whose (i, j)thentry equals the (j,i)thentry ofA. Thus theithrow ofATequals the (transpose of) theithcolumn ofA, and vice versa (for alli).A square matrixAis symmetric ifA= AT(i.e.,ithrow andithcolumn are the same vector).Example,
The identity(AB)T =BTAT(assuming that the productABexists) can be shown by considering the (i, j)thelement of (AB)T, i.e., the (j,i)thelement ofAB.This equals the dot product of thejthrow ofAand theithcolumn ofB; which is the same as the dot product of thethejthcolumn ofATand theithrow ofBT. This is just the (i, j)thelement ofBTAT.Row selections on anm×nmatrixAare accomplished by left-multiplyingAby the appropriate standard (row) unit vectors:(e(i))TA=ithrow ofAIfPis am × mpermutation matrix, thenPAis a permutation of the rows of A.Question: Ifn = mandAPputs the columns of A in a particular order, what shouldQbe so thatQAresults in the same reordering applied to row indices (instead of column indices)?
Example

If P is a permutation matrix, thenPTP=IThis is shown by considering the (i, j)thelement ofPTP, which equals the dot product of theithandjthcolumns ofP. Since the columns ofPare distinct standard unit vectors, it follows that the dot product equals unity ifi= j, zero otherwise