Discrete-Time State Models as Matrix Recurrences

From Update Rules to a State Vector

Many systems change in steps: inventory is recorded at the end of each week, or population counts are recorded once per breeding season. To model such a system, first decide which quantities must be known at time \(t\) to determine the state at time \(t+1\). Collect those quantities into a column vector \(\mathbf{x}_t\), called the state vector. A discrete-time linear state model with a fixed update rule has the form \(\mathbf{x}_{t+1}=\mathbf{A}\mathbf{x}_t\).

Here \(t\) indexes successive time steps, \(\mathbf{x}_t\in\mathbb{R}^n\), and \(\mathbf{A}\in\mathbb{R}^{n\times n}\) is the update matrix. The equation describes one update only: multiply the current state column by \(\mathbf{A}\) to obtain the next state column. The matrix describes the rules; the vector describes the system at a particular time.

Writing down this equation is not just a matter of placing numbers in a matrix. The entries must match the meanings and order of the chosen state variables. If \(\mathbf{x}_t=(x_{1,t},\dots,x_{n,t})^{\top}\), then the \(i\)th component of the update is

Thus row \(i\) gives the rule for producing the next value of state variable \(i\). Column \(j\) records how the current value of state variable \(j\) contributes to all components of the next state. This row-and-column interpretation follows from using column vectors.

Why a Linear Update Has a Matrix

The connection between a linear rule and a matrix is exact. A linear map \(F:\mathbb{R}^n\to\mathbb{R}^n\) is a function satisfying \(F(a\mathbf{u}+b\mathbf{v})=aF(\mathbf{u})+bF(\mathbf{v})\) for all \(\mathbf{u},\mathbf{v}\in\mathbb{R}^n\) and \(a,b\in\mathbb{R}\). Its matrix has one column for each standard basis vector. The next result proves that this construction always represents the rule and determines the matrix uniquely.

Proof. Let \(\mathbf{e}_1,\dots,\mathbf{e}_n\) be the standard basis of \(\mathbb{R}^n\), and form \(\mathbf{A}\) by placing \(F(\mathbf{e}_1),\dots,F(\mathbf{e}_n)\) as its columns. Every \(\mathbf{x}=(x_1,\dots,x_n)^{\top}\) has the expression \(\mathbf{x}=\sum_{j=1}^n x_j\mathbf{e}_j\). By linearity, \(F(\mathbf{x})=\sum_{j=1}^n x_jF(\mathbf{e}_j)=\mathbf{A}\mathbf{x}\), proving existence. If another matrix \(\mathbf{B}\) satisfies \(F(\mathbf{x})=\mathbf{B}\mathbf{x}\) for every \(\mathbf{x}\), then its \(j\)th column is \(\mathbf{B}\mathbf{e}_j=F(\mathbf{e}_j)\). Every column of \(\mathbf{B}\) therefore equals the corresponding column of \(\mathbf{A}\), so \(\mathbf{B}=\mathbf{A}\). \(\square\)

In applications, it is often more useful to start with component rules than with an abstract map. The same principle applies: express each next-state component as a linear combination of the current components, then place the coefficients into the corresponding row. The first worked example uses transfers between two inventory locations.

Columns Describe the Source Variables

For a state vector of counts or quantities, column \(j\) describes the contributions made by the \(j\)th current-state component to the next state. This perspective is especially useful for age-structured populations. Each column can be read as the next-period contributions associated with one current age class.

The model is a simplified rule, not a claim that every real population has these rates. In a model with fractional rates, the state components may represent expected counts or aggregate quantities rather than literal whole-number counts. The matrix still records the assumed linear update.

Choose State Variables That Determine the Next Step

A state vector must contain enough information to determine the next state under the proposed rules. If a future update depends on a quantity that the state omits, then the listed variables do not form a sufficient state for that rule. For example, current stock alone may not determine next period's stock when some replenishment is already in transit. Include the order pipeline as another component.

A Useful Check: Conserved Totals

In the warehouse-transfer model, total stock is preserved. This can be checked directly from the matrix: each column sums to \(1\). The following result explains why that check works for any state, not only the example's particular numbers.

Proof. The \(i\)th next-state component is \(x_{i,t+1}=\sum_{j=1}^n a_{ij}x_{j,t}\). Summing over \(i\) and collecting the terms multiplying each current component gives \(\sum_{i=1}^n x_{i,t+1}=\sum_{i=1}^n\sum_{j=1}^n a_{ij}x_{j,t}=\sum_{j=1}^n(\sum_{i=1}^n a_{ij})x_{j,t}\). By the hypothesis, every inner column sum \(\sum_{i=1}^n a_{ij}\) equals \(1\), so the final expression is \(\sum_{j=1}^n x_{j,t}\), as claimed. \(\square\)

For the inventory-transfer matrix, the column sums are \(\frac34+\frac14=1\) and \(\frac13+\frac23=1\). The proposition therefore guarantees conservation for any starting inventory vector. The population matrix does not have this property: its column sums are \(1\) and \(3\), reflecting the model's production of new juveniles.

Common Pitfalls and External Inputs

Using the transpose by mistake. Because state vectors are columns, the update is \(\mathbf{A}\mathbf{x}_t\), not \(\mathbf{A}^{\top}\mathbf{x}_t\). For the warehouse model, the incorrect transpose calculation for \((20,6)^{\top}\) would be

which does not give the stated warehouse rules or preserve the total stock. The correct product uses the original matrix and gives \((17,9)^{\top}\). Check the meaning of each row against the update rule before multiplying.

A constant external input is not part of a homogeneous linear update. For instance, a one-variable rule \(I_{t+1}=I_t-d\), with fixed \(d\ne0\), cannot be written as \(I_{t+1}=aI_t\) for all \(I_t\): at \(I_t=0\), the first rule gives \(-d\), while every linear rule \(aI_t\) gives \(0\). One way to represent a constant term is to add a fixed coordinate. With \(\mathbf{z}_t=(I_t,1)^{\top}\), the same update becomes

The added coordinate carries the constant input through the matrix product. Before using a recurrence, check whether all next-state terms are proportional to the current state variables or whether a fixed input must also be represented.