Proving T is Linear in Mnn: A + A^T

[chantella28]

Another question that I'm a little bit stumped on...

Define T: Mnn -> Mnn by T(A) = A + A^T. Prove that T is linear

i know that with T(A) = A^T you can prove it by the equations (if A and B are arbitrary matrices in Mnn and c is a scalar):
T(A+B) = (A+B)^T = (A)^T + (B)^T = T(A) + T(B)
T(cA) = (cA)^T = cT(A)

but i don't know how to manipulate these equations for the A + A^T problem...

 

[kleinwolf]

do you mean T(A)=A+A^T(A) ? (there's a redundant symbol describing not the same things) or T(A)=A+transpose(A)...the first is almost surely not linear, the second should be accepted to be proven like : T(A+B)=A+B+trn(A+B)=A+B+trn(A)+trn(B)=A+trn(A)+B+trn(B)=T(A)+T(B) with your defintion...the same for scalar multiplication

 

[chantella28]

sorry, i think it means the transpose... cos in the question its A to the power of capital T...

 

[Galileo]

You have to show that T(A+B)=T(A)+T(B) and T(cA)=cT(A)
Since T(A)=A+A^T, what do you get if you let T act on A+B?

 

[chantella28]

i don't understand what happens to the transpose though

 

[Galileo]

Are there any rules or identities with the transpose you are familiar with?

 

[vincebs]

Use the fact (that you already know from your original post) that A^T is linear to prove that A + A^T is linear.

 

[chantella28]

the transpose rules i know of are (A+B)transpose = Atranspose + Btranspose; (cA)transpose = cAtranspose; (AB)transpose = BtransposeAtranspose; and (Atranspose)transpose = A

but i still don't understand how to go about manipulating the addition and scalar multiplication formulas to prove that T(A)=A+Atranspose is linear

 

[Galileo]

You know what T does to an arbitrary matrix Q right? T(Q)=Q+Q^T.
So what you have to show is that for any two matrices A,B we have T(A+B)=T(A)+T(B)

So let Q=A+B. Then T(A+B)=(A+B)+(A+B)^T.
We also have T(cA)=(cA)+(cA)^T.

So are these equal to T(A)+T(B) and cT(A) respectively?