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Re: minors and cofactors

From: A. Scottedward Hodel
Subject: Re: minors and cofactors
Date: Mon, 21 Sep 1998 06:50:55 -0500

>> Calculate the SVD  A=U*S*V'.
>> Then det(A)=det(U)*det(S)*det(V')
>> and  inv(A)=V*inv(S)*U'.
>> Hence  inv(A)*det(A) = (det(V')*V) * (det(S)*inv(S)) * (det(U)*U')
>> The middle factor can be calculated by replacing each diagonal term
>> be the product of the others.
>just to throw my two cents in...
>note that U and V in the SVD are *unitary* matrices and hence have
>det(U)=det(V)=1.  S is diagonal hence the determinant is trivial to

    c = cos(pi/4), s =cos(pi/4).
    A = [c s ; -s c],   B = [c -s ; -s -c]
    A'A = B'B = I
    det(A) = 1, det(B) = -1.

If one considers the case of unitary matrices (complex valued
matrices with inv(A) = A', complex conj. transp, the determinant
can be any number on the complex unit circle.  Real orthogonal
matrices can have a determinant of +/- 1, as shown above.

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