Question 5.E.8.18: Let η be an arbitrary scalar, and let c = (1 η η² ... η^2n−1......
Let η be an arbitrary scalar, and let
c = \begin{pmatrix}1\\η\\η^2\\\vdots\\η^{2n−1}\end{pmatrix} and a = \begin{pmatrix}α_0\\α_1\\\vdots\\α_{n−1}\end{pmatrix}.Prove that c^T (a \odot a) = (c^T \hat{a})^2.
Step-by-Step
We know from (5.8.11) that if p(x) = \sum_{k=0}^{n-1} α_kx^k, then
p(x)q(x) = \sum\limits_{k=0}^{2n-2} \left[\sum\limits_{j=0}^{k}α_j β_{k−j}\right] x^k = \sum\limits_{k=0}^{2n-2} \left[a \odot b\right]_k x^k (5.8.11)
p^2 (x) = \sum\limits_{k=0}^{2n−2} [a \odot a]_k x^k.The last component of a \odot a is zero, so we can write
c^T (a \odot a) = \sum\limits_{k=0}^{2n−2} [a \odot a]_k η^k = p^2 (η) = \left(\sum\limits_{k=0}^{n−1}α_kη^k\right)^2 = \left(c^T \hat{a}\right)^2.Related Answered Questions
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