Question

A is a square matrix, show C=1/2(A+A^T) is symmetric

Answer 1

\[C^{T} = \left(\frac{1}{2}(A+A^{T})\right)^{T}= \frac{1}{2}(A+A^{T})^{T}=\frac{1}{2}((A^{T})^{T}+A^{T})\]
\[=\frac{1}{2}(A+A^{T})=C\]

Answer 2

make sense?

Answer 3

yup

Answer 4

Sometimes with this stuff, you just plow through it using the defs and it all comes out in the wash.

Answer 5

now even if (A+B)^T isn't (B^T + A^T), A+B = B+A, addition is commutative with square matrices.

Answer 6

so really it should be...
\[C^{T}=\left(\frac{1}{2}(A+A^{T})\right)^{T}=\frac{1}{2}(A+A^{T})^{T}=\frac{1}{2}(A^{T}+(A^{T})^{T})\]
\[=\frac{1}{2}(A^{T}+A)=\frac{1}{2}(A+A^{T})=C\]

Answer 7

@Aenn88 you awake? Do you follow what i did?

Answer 8

yes it does make sense. thank you for spelling it out!