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Determinant of Orthogonal

by salman213
Tags: determinant, orthogonal
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salman213
#1
Dec10-07, 07:08 PM
P: 303
Hi I had a final today and one of the questions was

find all the possible values of det Q if Q is a orthogonal matrix

I m still wondering how would I do this? Any ideas?
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cristo
#2
Dec10-07, 07:15 PM
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What is the definition of an orthogonal matrix?
salman213
#3
Dec10-07, 07:55 PM
P: 303
well i guess the vectors which make up the matrix are orthogonal and so have a dot product of 0?

and the transpose of an orthogonal matrix is its inverse


but im not sure how to use this to find out all values of the determinant

cristo
#4
Dec10-07, 08:01 PM
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Determinant of Orthogonal

Ok, so you know the transpose of an orthogonal matrix is its inverse. So, we have [itex]M^TM=I[/itex]. Now, let's take the determinant of this; [itex]det(M^TM)=det(I)[/itex]. I presume you know what the right hand side is equal to. Now, what can one say about the relationship between the determinant of a matrix, and the determinant of its transpose?
salman213
#5
Dec15-07, 07:53 PM
P: 303
but how is the determinant of(M^TM) = det(M)

if M is a orthogonal matrix




by the way since you said det (i) its 1..right?

and I do know the det(M^t) = det (M)

but det (M^tM) = 1 and im not understanding how that is = det (M)
cristo
#6
Dec15-07, 07:55 PM
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Quote Quote by salman213 View Post
and I do know the det(M^t) = det (M)

but det (M^tM) = 1 and im not understanding how that is = det (M)
Right, so putting these two facts together we have det(M2)=1. Can you find det(M) from this expression?
mjsd
#7
Dec15-07, 08:06 PM
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hmmm...salman213 perhaps this is the theorem you want

det(AB) = det(A)det(B)
salman213
#8
Dec16-07, 03:09 PM
P: 303
oh okk..cool..thanks


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