23.9.2011
Jaroslav Kautsky (Flinders University, Australia)
Generalized Toeplitz-Pascal matrices and bi-diagonal LU factorization
Abstract:
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A class of matrices which are the Hadamard product of a
fixed lower triangular generating matrix P and any
Toeplitz matrix is studied. Conditions on P which lead
to the class being closed under matrix multiplication are
derived and such generating matrices are fully characterized.
This generalizes the special cases when either P itself is Toeplitz
or P is the Pascal triangle but the Toeplitz matrix is
generated by the vector of powers.
Explicit formulae for inverses are derived and commutativity
of products within each class proven.
Examples using generalized Pascal triangles are shown.
The known factorization of the Pascal triangle/power Toeplitz
matrix into a product of bi-diagonal matrices is generalized
for any matrix with certain non-vanishing minors.
Motivation for the study of these properties came from the
bad condition of the Pascal triangle matrix which causes
difficulties in the evaluation of blur invariants of higher
degrees. That problem is, however, unresolved.