Exam 16: Markov Processes

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A fundamental matrix, denoted as N, is used in the field of differential equations, specifically in the context of linear homogeneous systems of first-order differential equations. It is a matrix solution to a system of linear differential equations and plays a crucial role in understanding the behavior of the solutions to these systems.

In the context of a linear system of differential equations, the system can be written in matrix form as:

dx/dt = A(t) x,

where x is a vector of functions representing the system's state, A(t) is a time-dependent matrix, and t represents time.

The fundamental matrix N(t) is a square matrix whose columns are linearly independent solutions to the system. It satisfies the equation:

dN/dt = A(t) N,

with the initial condition that N(t0) is an invertible matrix, often chosen to be the identity matrix I for simplicity.

The fundamental matrix is important because once it is found, any solution to the system can be expressed as a linear combination of the columns of N. If x(t) is a solution to the system, then there exists a constant vector c such that:

x(t) = N(t) c.

The computation of the fundamental matrix N can be done in several ways, depending on the nature of the matrix A(t):

1. **Constant Coefficients:** If A(t) is a constant matrix, the fundamental matrix can be computed using the matrix exponential, N(t) = exp(A t), where exp denotes the matrix exponential function.

2. **Variation of Parameters:** For non-constant A(t), one can use the method of variation of parameters. This involves guessing a form for N(t) and then determining the unknown functions by plugging N(t) into the differential equation and solving the resulting equations.

3. **Series Solutions:** When A(t) has a specific form, such as periodicity or other properties, series solutions can be used to construct N(t) by assuming a series form for the solutions and determining the coefficients.

4. **Numerical Methods:** When an analytical solution is not feasible, numerical methods such as the Runge-Kutta method can be used to approximate N(t) by discretizing the time variable and iteratively solving the system of equations.

5. **Diagonalization or Jordan Normal Form:** If A is diagonalizable, N(t) can be computed by diagonalizing A and then taking the matrix exponential of the diagonal matrix. If A is not diagonalizable, one can use its Jordan normal form to compute the matrix exponential and hence the fundamental matrix.

The computation of the fundamental matrix is a fundamental tool in the analysis of linear systems of differential equations, as it provides a way to understand the complete solution space of the system.