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Expert: - 2/18/2009


Question
1a). Let P be an invertible nxn matrix. Prove that
(P^-1)^T= (P^T)^-1
(You may use the fact that (P^T)^T =P.)

b) Let Mn denote the set nxn square matrices with real entries. We define a relation R on this set as follows: M R N  if and only if there exists and invertible P such that M = P^(T) NP. Prove that p is an equivalence relation.




2 a) Find the roots of z^5 = i for z complex.

b) Setting X= cos(ipi/5) + isin(ipi/5),show that the sum of the five roots equals

X(1-X^10)/(1-X^2)
Hence deduce that the sum of the roots is zero. (Use De Moivre's Formula.)


Thank you very much for your help.

Answer
According to en.wikipedia.org/wiki/Transpose , we know that
(A^T)^-1 = (A^-1)^T, so that's in the paper as #8.

It is also found in fourier.eng.hmc.edu/e161/lectures/algebra/node2.html,
down at the bottom.

If it is thought about, it would seem reasonable the
inverse of the transform would be the same as
the transform of the ineverse.

Here is a good place about equivalance relations:
www.ee.ic.ac.uk/hp/staff/dmb/matrix/relation.html .

2a) To find the 5th root of i, think of a graph with the reals on
the x axis and the imaginary on the y axis.  Convert i to polar
coordinates.  It would be (1,π/2).  To find the 5th root,
take the 5th root of 1 { which is still 1 } and divide that angle
by 5.  You must also add 2π/5, 4π/5, 6π/5, and 8π/5 so that you
have 5 5th roots to the number.

b) The sum of the 5 roots would be the sum(i = 0 to 4) of
cos(iπ/5)+isin(iπ/5).

It can be seen the the roots over the real line would be just the
opposite of the roots beneath the real line.  It can also be seen
that the sum of the roots to the left of the imaginary line would
be the same as the sum of the roots to the right of the imaginary
line.

I found a very good definition of De Moivre's Formula at
en.wikipedia.org/wiki/De_Moivre's_formula .

DeMoivre's Identity can be found in
mathworld.wolfram.com/deMoivresIdentity.html .

His major formula was that (cosx + isinx)^n = cos(nx) + isin(nx),
which is sometimes referred to as cis(x).

I wish I could be of more assistance, but I haven't dealt with
complex roots that much yet.  The only thing I know about them
right now is can be found in what I've written.

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