Question: A Toeplitz matrix is an n n matrix A = (a ij ) such that a ij = a i - 1 . j

A Toeplitz matrix is an n × n matrix A = (aij) such that aij = ai-1.j-1 for i = 2, 3, . . . , n and j = 2, 3 , . . . , n.

a. Is the sum of two Toeplitz matrices necessarily Toeplitz? What about the product?

b. Describe how to represent a Toeplitz matrix so that you can add two n × n Toeplitz matrices in O(n) time.

c. Give an O(n lg n)-time algorithm for multiplying an n × n Toeplitz matrix by a vector of length n. Use your representation from part (b).

d. Give an efficient algorithm for multiplying two n × n Toeplitz matrices. Analyze its running time.

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