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Some Linear Algebra questions

Hey guys, I have absolutely no idea where to go with the following questions and any help/direction would be immensely appreciated:

Q1 If "A" is an orthogonal matrix, find the value(s) for c, if possible, so that cA is also orthogonal.

(For this one, I get a feeling it involves working something like cAcAT=IcAcA^T = I, but I'm not certain.

Q2 Construct an orthonormal basis for the subspace W of R4R^4 spanned by {(1,1,0,0), (2,-1,0,1), (3, -3, 0, -2), (1, -2, 0, -3)}

Q3 If "A" is any n x n matrix that is diagonalisable, prove that A−1A^{-1} is diagonalisable.
Reply 1
Avatar for nota bene
nota bene
15
A is ortogonal i.e. A.A^T=A^T.A=I for cA to be orthogonal you have that (cA).(cA)^T=I (before you go any further you can note it's trivially true for c=1 and clearly not true for c=0 as the zero matrix is not equal to the identity matrix... however this will fall out at a later stage anyway)

Now, c(A^T)=(cA)^T (convince yourself of this!). Also, pre- or postmultiplication with constants doesn't matter i.e. kA=Ak so thus you can say:

Do not read until you've thought about it



Now you should be able to finish it off.
*Hopes he's made no silly mistakes*

edit: As for question 2 I'm convinced there will be examples of that in your notes, it is bookwork.
Reply 2
Avatar for nota bene
nota bene
15
A diagonalisable means there is a P (invertible) s.t. P.A.P^(-1)=D where D is a diagonal matrix. So A=P^(-1).D.P and thus A^(-1)=(P^(-1).D.P)^(-1), can you finish this off?

Hint



edit: You actually don't need to do A=P^(-1).D.P, it is better to leave it as P.A.P^(-1)=D and invert that.
Reply 3
Avatar for final_alert
final_alert
OP
Sweet. Thanks guys! I had thought about it between checking the threads and worked it the way you both have.

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