The Hadamard matrices${\mathit{H}}_{\mathbf{0}}\mathbf{,}{\mathit{H}}_{\mathbf{1}}\mathbf{,}{\mathit{H}}_{\mathbf{2}}\mathbf{,}\mathbf{\dots }$ are defined as follows:

• • ${\mathit{H}}_{\mathbf{0}}$ is the $\mathbf{1}\mathbf{\times }\mathbf{1}\mathbf{}\mathit{m}\mathit{a}\mathit{t}\mathit{r}\mathit{i}\mathit{x}\mathbf{}\mathbf{\left[}\mathbf{1}\mathbf{\right]}\mathbf{}$
• For $\mathit{k}\mathbf{>}\mathbf{0}\mathbf{,}\mathbf{}{\mathit{H}}_{\mathbf{k}}\mathit{i}\mathit{s}\mathbf{}\mathit{t}\mathit{h}\mathit{e}\mathbf{}{\mathbf{2}}^{\mathbf{k}}\mathbf{\times }{\mathbf{2}}^{\mathbf{k}}$ matrix

localid="1658916810283" ${\mathit{H}}_{\mathbf{k}}\mathbf{=}\mathbf{\left[}\frac{{\mathbf{H}}_{\mathbf{k}\mathbf{-}\mathbf{1}}\mathbf{|}{\mathbf{H}}_{\mathbf{k}\mathbf{-}\mathbf{1}}}{{\mathbf{H}}_{\mathbf{k}\mathbf{-}\mathbf{1}}\mathbf{|}\mathbf{-}{\mathbf{H}}_{\mathbf{k}\mathbf{-}\mathbf{1}}}\mathbf{\right]}$

Show that if $\mathit{\upsilon }$ is a column vector of lengthlocalid="1658916598888" $\mathit{n}\mathbf{=}{\mathbf{2}}^{\mathbf{k}}$, then the matrix-vector product localid="1658916618774" ${\mathit{H}}_{\mathbf{k}}\mathit{v}$can be calculated using localid="1658916637767" $\mathit{O}\mathbf{(}\mathit{n}\mathbf{}\mathit{l}\mathit{o}\mathit{g}\mathit{n}\mathbf{}\mathbf{)}\mathbf{}$ operations. Assume that all the numbers involved are small enough that basic arithmetic operations like addition and multiplication take unit time.

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