I have a problem and it seems that my linear algebra classes are too long ago. I have a matrix $M \in \mathbb{C}^{a x a}$. Additionally, I have a set of matrices, that build a basis $B={b_i}$ of that matrix space.
I am now interested in only in only one of the coefficients, of $M$, represented in the basis $B$. To be more clear $M=\sum a_ib_i$ and only need $a_1$. Is there an elegant solution to this?
Thanks in advance, glostas
PS: If it helps: all matrices are hermition.
There is, at the moment, no need to think about $M$ and the $b_i$ as matrices. $M$ and the $b_i$ are, for our intents and purposes, complex vectors in dimension $a^2$, expressed in the standard basis. I will rearrange them to be column vectors for notational convenience.
We want the first coordinate of $M$ when it is written in the base given by the $b_i$. This is given by the first entry in $B^{-1}M$, where $B$ is the $a^2\times a^2$ matrix with the $b_i$ as columns. Unless the $b_i$ happen to be orthogonal, or something similarly nice, there is probably no easy and accessible way around the calculation of $B^{-1}$.
If the $b_i$ happen to be orthogonal, or at least if $b_1$ is orthogonal to the rest, then we can get away with just $\frac1{\|b_1\|^2}b_1{}^TM$.