These are quick scribes from this really nice Lecture here by Maria Schuld.
We consider a measurement matrix $M$ which is a diagonal of measurement values and the corresponding probability assignments $p$ to each.
The expectation of such a random variable can now be represented in a quadratic form vector product such that for a vector $q$, $q_i^2 = p_i$ as
Quantum theory revolves around computing expectation of measurements and these ideas from classical linear algebra are extended in a general form as
$M$ in the most general case (can have non-diagonal elements as well) is a Hermitian matrix with eigen values equal to the measurements.
Playing in this world is all about manipulating $\psi$ via unitary matrices $U$.
Different quantum computing models are polynomially equivalent.