2022-07-20

Can I simplify the following vector?
Let a and b be $n×1$ vectors and ${M}_{1},{M}_{2},...,{M}_{n}$ be $n×n$ matrices.
Is there a way to write the following vector:
$v=\left[\begin{array}{c}{a}^{T}{M}_{1}b\\ {a}^{T}{M}_{2}b\\ ⋮\\ {a}^{T}{M}_{n}b\end{array}\right]$
in such a way that reads $v={a}^{T}{M}_{new}b$, where ${M}_{new}$ is just a manipulation of the ${M}_{i}$ matrices?
I know I can do:
$v=\left[\begin{array}{c}{a}^{T}{M}_{1}\\ {a}^{T}{M}_{2}\\ ⋮\\ {a}^{T}{M}_{n}\end{array}\right]b$
But then I cannot get the vector a "out of the brackets".

sviudes7w

Expert

You could write
$v=\left(\begin{array}{c}{a}^{T}\\ & \ddots & \\ & & {a}^{T}\end{array}\right)\left(\begin{array}{c}{M}_{1}\\ ⋮\\ {M}_{n}\end{array}\right)b.$
If you like, you could write the matrix on the left compactly using the fact that
$\left(\begin{array}{c}{a}^{T}\\ & \ddots & \\ & & {a}^{T}\end{array}\right)={a}^{T}\otimes I,$
where I denotes a size n identity matrix and ⊗ denotes the Kronecker product.

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