Can someone help me about skew symmetric?

[JerryKelly]

Let A be an nxn skew symmetric mx.(A^T=-A).
i) Show that if X is a vector in R^n then (X,AX)=0
ii) Show that 0 is the only possible eigenvalue of A
iii)Show that A^2 is symmetric
iv)Show that every eigenvalue of A^2 is nonpositive.
v)Show that if X is an eigenvector of A^2 , then so is AX
vi)With X as in v), show that the subspace W spanned by X and AX is an A-subspace.
vii)Assuming AX not= 0. let U=X/||X||, V=AX/||AX||. Show that AU=(AU,V)V and that AV=-(AU,V)U.
ix) Show if U,V,U3 , ...,Un is an orthonormal basis of R^n, then the mx of a reletive to this basis has the form (0,b,0,0...0;-b,0,0,0...0;0,0,0,a,0...0;0,0,-a,0,0...0;C some format to the end), where C is skew symmetric (n-2)x(n-2) mx.


Thanks!

 

[Galileo]

This smells like homework. What have you tried on these questions so far?

 

[matt grime]

i) follows from the definition of transpose and the properties of inner products, what do you know about them?

 

[JerryKelly]

Yes,it is homework question. So far, I just have a idea for part iv), and I have no idea for the rest of them. For part iv), i can use (AX,X)=(X,-AX)=-A(X,X). Except this one, I have no idea how to do the rest of them. Could you give me some help,please? Thanks!

This smells like homework. What have you tried on these questions so far?

 

[matt grime]

I think you need to relearn some basics, or correct some typos.

How can (x,-Ax)=-A(x,x)?

The left hand side is a scalar, the right hand side is a matrix (times a scalar)

Remember the definition of transpose is made to satisfy

$$(Mx,y)=(x,M^Ty)$$

and that pretty much is all you need, in fact it solves pretty much everything (along with the definition of eigenvalue/vector)

 

[JerryKelly]

Could you solve the last two parts by using this definition for me,please? I have no any idea for the last two parts. Thanks!

I think you need to relearn some basics, or correct some typos.

How can (x,-Ax)=-A(x,x)?

The left hand side is a scalar, the right hand side is a matrix (times a scalar)

Remember the definition of transpose is made to satisfy

$$(Mx,y)=(x,M^Ty)$$

and that pretty much is all you need, in fact it solves pretty much everything (along with the definition of eigenvalue/vector)