document.write( "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
\n" ); document.write( "(a)show that H is symmetric
\n" ); document.write( "(b)show that H^-1=H^T \n" ); document.write( "

Algebra.Com's Answer #677951 by rothauserc(4718) $\"\"$ $\"About$

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Note the general formula for householder is H = I(n) - 2WW^T/((W^T)W) = I(n) + beta(WW^T)
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\n" ); document.write( "(a) H is symmetric then we need to show H^T = H
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\n" ); document.write( "H = I(n) - 2WW^T
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\n" ); document.write( "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
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\n" ); document.write( "I used the basic properties of a transposed matrix
\n" ); document.write( "(1) for alpha a scaler (aA)^T = aA^T
\n" ); document.write( "(2) (A+B)^T = A^T + B^T
\n" ); document.write( "(3) (AB)^T = (B^T)A^T
\n" ); document.write( "(4) (A^T)^T = A
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\n" ); document.write( "(b) show H^(-1) = H^T
\n" ); document.write( "from definition HH^(-1) = H^(-1)H = I(n)
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\n" ); document.write( "H is orthogonal, namely H(H^T) = I(n), since H is symmetric (H^T)H = I(n)
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\n" ); document.write( "To show H is orthogonal multiply (I(n) - 2WW^T) by itself and factor it out
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