Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  show that every $$n\,\times \,n$$n×n matrix is generically a product of $$\lfloor n/2 \rfloor + 1$$?n/2?+1 Toeplitz matrices and always a product of at most $$2n+5$$2n+5 Toeplitz matrices. The same result holds true if the word `Toeplitz' is replaced by `Hankel,' and the generic bound $$\lfloor n/2 \rfloor + 1$$?n/2?+1 is sharp.  will see that these decompositions into Toeplitz or Hankel factors are unusual:  may not, in general, replace the subspace of Toeplitz or Hankel matrices by an arbitrary $$(2n-1)$$(2n-1)-dimensional subspace of $${n\,\times \,n}$$n×n matrices. Furthermore, such decompositions do not exist if  require the factors to be symmetric Toeplitz or persymmetric Hankel, even if  allow an infinite number of factors.
['Ke Ye', 'Lek-Heng Lim']
I am writing a research paper on a idea Every Matrix is a Product of Toeplitz Matrices
[{"from": "human", "value": "I am writing a research paper on a idea Every Matrix is a Product of Toeplitz Matrices"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  show that every $$n\\,\\times \\,n$$n\u00d7n matrix is generically a product of $$\\lfloor n/2 \\rfloor + 1$$?n/2?+1 Toeplitz matrices and always a product of at most $$2n+5$$2n+5 Toeplitz matrices. The same result holds true if the word `Toeplitz' is replaced by `Hankel,' and the generic bound $$\\lfloor n/2 \\rfloor + 1$$?n/2?+1 is sharp.  will see that these decompositions into Toeplitz or Hankel factors are unusual:  may not, in general, replace the subspace of Toeplitz or Hankel matrices by an arbitrary $$(2n-1)$$(2n-1)-dimensional subspace of $${n\\,\\times \\,n}$$n\u00d7n matrices. Furthermore, such decompositions do not exist if  require the factors to be symmetric Toeplitz or persymmetric Hankel, even if  allow an infinite number of factors."}]