# Evolution of systems

### Vectors

Consider a purely abstract n-dimensional vector space $\mathbb{V}$, over a field $\mathbb{F}$ (either the reals or the complex numbers). We can represent the vectors as columns and elements of the dual space, $\mathbb{V}^*$, as rows, $$\left [ \begin{matrix} a_1 \\ \vdots \\ a_n \end{matrix} \right ] \in \mathbb{V}$$ and $$[ a_1 \cdots a_n ] \in \mathbb{V}^*$$ where $a_i \in \mathbb{F}$.

We define the inner product (dot product in $\mathbb{R}^3$) by:

$$[ b_1 \cdots b_n ] \left [ \begin{matrix} a_1 \\ \vdots \\ a_n \end{matrix} \right ] = b_1 a_1+... +b_n a_n = \sum_{i = 1}^{n} b_i a_i = b_i a_i$$ using the Einstein summation convention (ESC).

### Matrices

We can consider an n-dimensional square matrix $M$ as a linear operator over a n-dimensional vector space, $\mathbb{V}$. That is, $M:\mathbb{V} \to \mathbb{V}$, such that $$\forall \alpha, \beta \in \mathbb{F} \text{ and } \forall \ket{a},\ket{b} \in \mathbb{V}\text{ we have } M(\alpha \ket{a} + \beta \ket{b})=\alpha M\ket{a} + \beta M\ket{b}$$

In column form, $$\left [ \begin{matrix}m_{11} & \cdots & m_{1n}\\ \vdots & & \vdots\\ m_{n1} & \cdots & m_{nn}\end{matrix} \right ]\left [ \begin{matrix} a_1 \\ \vdots \\ a_n \end{matrix} \right ]=\left [ \begin{matrix} m_{1i} a_i \\ \vdots \\ m_{ni} \ a_i \end{matrix} \right ]$$

The same matrix can operate on the dual space as well:

$$\left [ \begin{matrix} b_1 \cdots b_n \end{matrix} \right ]\left [ \begin{matrix} m_{11} & \cdots & m_{1n} \\ \vdots & & \vdots \\ m_{n1} & \cdots & m_{nn} \end{matrix} \right ]=\left [ \begin{matrix} b_i m_{i1} \cdots b_i m_{in} \end{matrix} \right ]$$

Nb. We have used the Einstein Summation Convention in both of the resulting vectors.

### Time evolution

Figure 1.3 - A simple rule for updating

Consider the rule shown in figure 1.3. The states $\{\ket{1},\ket{2},\ket{3},\ket{4}\}$ could be represented by the column vectors $$\left\{\:\left [ \begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix} \right ],\left [ \begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix} \right ],\left [ \begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix} \right ],\left [ \begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix} \right ]\:\right\}$$

The rule for updating the system can be represented by the linear operator $$M =\left [ \begin{matrix}0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\end{matrix} \right ]$$.

Then, $M\ket{1}=\ket{2}, M\ket{2}=\ket{3}, M\ket{3}=\ket{4} \text{ and } M\ket{4}=\ket{1}$.

We can consider the time evolution of a system as successive applications of a linear operator on a vector state. Given a state $v$, we could write $v_0 = v, v_1 = Mv, v_2 = M(Mv) = M^2 v, \dots, v_n = M^n v$, where $M^n$ can be calculated using matrix multiplication.