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Question 2.7.19: Suppose R is rectangular (m by n) and A is symmetric (m by m...

Suppose R is rectangular (m by n) and A is symmetric (m by m).
(a) Transpose {R}^{T}AR to show its symmetry. What shape is this matrix?
(b) Show why {R}^{T}R has no negative numbers on its diagonal.

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(a) The transpose of {R}^{T}AR is {R}^{T}{A}^{T}{R}^{TT} = {R}^{T}AR = n by n when {A}^{T} = A (any m by n matrix R)

(b) ({R}^{T}R)_{jj} = (column j of R) \cdot (column j of R) = (length squared of column j ) \ge 0.

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