[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
## 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*
>*compute.*
>
Careful.
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.