SOLUTION: let W be an nx1 matrix such that W^T*W=1.The nxn matrix H=In-2WW^T is called a householder matrix (a)show that H is symmetric (b)show that H^-1=H^T

Question 1062934: let W be an nx1 matrix such that W^T*W=1.The nxn matrix H=In-2WW^T is called a householder matrix
(a)show that H is symmetric
(b)show that H^-1=H^T
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
Note the general formula for householder is H = I(n) - 2WW^T/((W^T)W) = I(n) + beta(WW^T)
:
(a) H is symmetric then we need to show H^T = H
:
H = I(n) - 2WW^T
:
H^T = (I(n) -2WW^T)^T = I(n)^T -2(WW^T)^T = I(n) -2(W^T)^TW^T = I -2WW^T = H
:
I used the basic properties of a transposed matrix
(1) for alpha a scaler (aA)^T = aA^T
(2) (A+B)^T = A^T + B^T
(3) (AB)^T = (B^T)A^T
(4) (A^T)^T = A
:
(b) show H^(-1) = H^T
from definition HH^(-1) = H^(-1)H = I(n)
:
H is orthogonal, namely H(H^T) = I(n), since H is symmetric (H^T)H = I(n)
:
To show H is orthogonal multiply (I(n) - 2WW^T) by itself and factor it out
:

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