## 2 Linear Modes  
Modal concepts are classical in linear system theory, e.g., see reference [11]. Consider the general class of linear time-invariant ordinary differential equation (ODE) models, with inputs and outputs specified integral to model \(M\), as in Figure 1 and Equation (1):  
\[\frac{d\mathbf{q}(t)}{dt}=K\mathbf{q}(t)+B\mathbf{u}(t),\quad\mathbf{y}(t)=C\mathbf{q}(t),\quad t\geq 0 \tag{1}\]  
Symbols \(\mathbf{q}\), \(\mathbf{u}\) and \(\mathbf{y}\) are \(n\)-, \(r\)- and \(m\)-vectors of time-varying state variables (system), inputs and outputs, respectively. \(K\), \(B\) and \(C\) are \(n\) by \(n\) system, \(n\) by \(r\) input and \(m\) by \(n\) output matrices of constants, respectively. Multicompartmental (MC) models, defined graphically in Section 4 below, are a special case, with \(K\equiv\left[k_{ij}\right]\) having the properties \(k_{ij}\geq 0\) for \(i\
eq j\) and \(\sum_{i=0}^{n}k_{ij}=0\), \(j=1,\ldots,n\)[12; 13]. The component terms of the solution \(\mathbf{y}(t)=C\mathbf{q}(t)=Ce^{Kt}\mathbf{q}(0)\) of the homogeneous equation \(d\mathbf{q}/dt=K\mathbf{q}\), i.e., the response to initial conditions (ICs) \(\mathbf{q}(0)\) (input \(\mathbf{u}\equiv\mathbf{0}\)) are the **modes** of model (1). This _fundamental set_ of solution terms (modes) is \(n\) linearly independent exponential functions of the form \(t^{k}e^{\lambda_{i}t}\) (\(k=0,\,1,\,\ldots,\,n-1\) and \(j=1,\,2,\,\ldots,\,n\)), each satisfying \(d\mathbf{q}/dt=K\mathbf{q}\). The \(\lambda_{j}\) are _eigenvalues_ of \(K\). If they are distinct, the \(n\) modes are \(y_{1}=e^{\lambda_{1}t},\,\ldots,\,y_{n}=e^{\lambda_{n}t}\). If they are repeated, then for each \(\lambda_{i}\) of multiplicity \(n_{i}\) (i.e., \(n_{i}\) is the number of \(\lambda_{i}\)), there are \(n_{i}\) modes: \(e^{\lambda_{i}t},\,\,te^{\lambda_{i}t},\,\ldots,\,\,t^{n_{i}-1}e^{\lambda_{i}t}\), also totaling \(n\). Importantly, \(n\) is the minimum dynamical dimension of model (1), i.e., the model cannot be reduced to a smaller number of ODEs that completely characterize dynamical responses \(\mathbf{y}(t)=C\mathbf{q}(t)=Ce^{Kt}\mathbf{q}(0)\). This is paramount in tying the model with I-O data, as we see below. The eigenvalues \(\lambda_{j}\) can be real or complex numbers. When they are complex, the modes are _oscillatory_. We now give all of this a geometric interpretation to tie it to the discussion of nonlinear (NL) models and modes in Sections 10 and 11. Each mode actually has a **shape**, represented by an _eigenvector_ component \(v_{j}\), associated with the **frequency** (eigenvalue) component \(\lambda_{j}\), so it can be represented geometrically as a **two-dimensional** function \(c_{i}\mathbf{v}_{j}t^{k}e^{\lambda_{i}t}\) each with its own eigenvalue \(\lambda_{j}\) and (\(n\)-dimensional) eigenvector \(\mathbf{v}_{j}\). The intrinsic parameters  
Figure 1: Linear system model \(M\) block diagram representation with input \(u\) and output \(y\) (I-O data). of \(\mathbf{y}(t)=\mathbb{C}\mathbf{q}(t)=\mathbb{C}e^{Kt}\mathbf{q}(0)\) are then the eigenvalues and eigenvectors of \(K\), i.e., a set of \(2n\)_structural invariants_ of \(K\). **Structural invariants** are one of many smallest sets of parameters completely determining or characterizing model dynamics [14]. _Example 1: **Simplest Case--Distinct Eigenvalues**_  
The dynamical shape of the solution modes, for recognition purposes as well as math analysis, depends in part on relationships among the \(n\) eigenvalues \(\lambda_{1}\), \(\lambda_{2}\),..., \(\lambda_{n}\) of system matrix \(K\). If they are _distinct_ (none are repeated) and the corresponding eigenvectors are \(v_{1}\), \(v_{2}\),..., \(v_{n}\), the general solution of \(dq/dt=K\mathbf{q}\) can be written:  
\[\mathbf{q}(t)=\sum_{i=1}^{n}c_{i}\mathbf{v}_{i}e^{\lambda_{i}t} \tag{2}\]  
The \(c_{i}\) are chosen to match the _initial conditions_ (ICs) on \(\mathbf{q}(t)\).