Is $a^T M b = b^T M a$?

Question

I am trying to reproduce a proof and I'm stuck at one point where it looks like this :

$a^T M b + b^T M a = 2 a^T M b$

Therefore the only explanation I can get is this, if true :

is $a^T M b = b^T M a$ ? (Preferrably with an explanation or a linear algebra property)

Note : a and b are (dx1) vectors, and M is a (dxd) matrix.

Thanks for any help!

Edit: Thanks for the answers, in my case, M is a symmetric covariance matrix.

Answer 1

You have that $a^\intercal M b$ is a scalar, so $a^\intercal Mb=(a^\intercal Mb)^\intercal = b^\intercal M^\intercal a$. So the result is true if $M$ is symmetric.

Answer 2

If $M$ is symmetric, yes, because $(AB)^T=B^TA^T$ and $(A^T)^T=A$.

Answer 3

Yes $a^TMb$ and $b^TMa$ are scalars, thus $(a^TMb)^T=a^TMb$, but using the rule that for two vectors $c,d$, $(c^Td)^T=d^Tc^{TT}=d^Tc$. we have:

$(a^TMb)^T=b^T(a^TM)^T=b^T(M^Ta)=b^TMa$.