What is the equation for the fundamental matrix?
2 Answers
The fundamental matrix is used in the context of solving systems of linear differential equations. For a system of first-order linear ordinary differential equations represented as:
\[
\mathbf{y}' = \mathbf{A}(t) \mathbf{y}
\]
where \(\mathbf{y}\) is a vector of functions and \(\mathbf{A}(t)\) is a matrix of functions, the fundamental matrix \(\Phi(t)\) is a matrix solution to this system. The fundamental matrix satisfies the following:
\[
\Phi'(t) = \mathbf{A}(t) \Phi(t)
\]
with the initial condition:
\[
\Phi(t_0) = \mathbf{I}
\]
where \(\mathbf{I}\) is the identity matrix and \(t_0\) is the initial time. The columns of \(\Phi(t)\) consist of linearly independent solutions to the system of differential equations.
\[
\mathbf{y}' = \mathbf{A}(t) \mathbf{y}
\]
where \(\mathbf{y}\) is a vector of functions and \(\mathbf{A}(t)\) is a matrix of functions, the fundamental matrix \(\Phi(t)\) is a matrix solution to this system. The fundamental matrix satisfies the following:
\[
\Phi'(t) = \mathbf{A}(t) \Phi(t)
\]
with the initial condition:
\[
\Phi(t_0) = \mathbf{I}
\]
where \(\mathbf{I}\) is the identity matrix and \(t_0\) is the initial time. The columns of \(\Phi(t)\) consist of linearly independent solutions to the system of differential equations.
The fundamental matrix is a key concept in control theory and systems analysis, particularly when dealing with linear time-invariant (LTI) systems. It provides a way to solve systems of linear differential equations.
For a system described by the linear differential equation
\[ \frac{d\mathbf{x}}{dt} = \mathbf{A}(t)\mathbf{x}, \]
where \(\mathbf{A}(t)\) is an \(n \times n\) matrix of functions of \(t\), the fundamental matrix \(\Phi(t)\) is an \(n \times n\) matrix solution to this differential equation.
**Fundamental Matrix Equation:**
The fundamental matrix \(\Phi(t)\) is defined as a matrix whose columns are linearly independent solutions to the differential equation. If \(\Phi(t)\) is a fundamental matrix, it satisfies the following differential equation:
\[ \frac{d\Phi(t)}{dt} = \mathbf{A}(t)\Phi(t). \]
The initial condition for the fundamental matrix is:
\[ \Phi(0) = \mathbf{I}, \]
where \(\mathbf{I}\) is the \(n \times n\) identity matrix.
### Properties of the Fundamental Matrix:
1. **Invertibility:**
- The fundamental matrix \(\Phi(t)\) is always invertible for all \(t\) if \(\mathbf{A}(t)\) is continuous.
2. **Solution to Non-Homogeneous Systems:**
- For a non-homogeneous system of the form \(\frac{d\mathbf{x}}{dt} = \mathbf{A}(t)\mathbf{x} + \mathbf{b}(t)\), the general solution can be expressed as:
\[ \mathbf{x}(t) = \Phi(t) \mathbf{x}_0 + \Phi(t) \int_{t_0}^t \Phi^{-1}(\tau) \mathbf{b}(\tau) \, d\tau, \]
where \(\mathbf{x}_0\) is the solution to the homogeneous system.
3. **Matrix Exponential Relation:**
- In the case where \(\mathbf{A}(t)\) is constant (i.e., \(\mathbf{A}(t) = \mathbf{A}\)), the fundamental matrix solution is given by:
\[ \Phi(t) = e^{\mathbf{A}t}, \]
where \(e^{\mathbf{A}t}\) is the matrix exponential of \(\mathbf{A}t\).
In summary, the fundamental matrix \(\Phi(t)\) is a matrix solution to the linear differential system with initial condition \(\Phi(0) = \mathbf{I}\). It plays a crucial role in analyzing and solving systems of linear differential equations, especially when dealing with initial value problems and non-homogeneous terms.
For a system described by the linear differential equation
\[ \frac{d\mathbf{x}}{dt} = \mathbf{A}(t)\mathbf{x}, \]
where \(\mathbf{A}(t)\) is an \(n \times n\) matrix of functions of \(t\), the fundamental matrix \(\Phi(t)\) is an \(n \times n\) matrix solution to this differential equation.
**Fundamental Matrix Equation:**
The fundamental matrix \(\Phi(t)\) is defined as a matrix whose columns are linearly independent solutions to the differential equation. If \(\Phi(t)\) is a fundamental matrix, it satisfies the following differential equation:
\[ \frac{d\Phi(t)}{dt} = \mathbf{A}(t)\Phi(t). \]
The initial condition for the fundamental matrix is:
\[ \Phi(0) = \mathbf{I}, \]
where \(\mathbf{I}\) is the \(n \times n\) identity matrix.
### Properties of the Fundamental Matrix:
1. **Invertibility:**
- The fundamental matrix \(\Phi(t)\) is always invertible for all \(t\) if \(\mathbf{A}(t)\) is continuous.
2. **Solution to Non-Homogeneous Systems:**
- For a non-homogeneous system of the form \(\frac{d\mathbf{x}}{dt} = \mathbf{A}(t)\mathbf{x} + \mathbf{b}(t)\), the general solution can be expressed as:
\[ \mathbf{x}(t) = \Phi(t) \mathbf{x}_0 + \Phi(t) \int_{t_0}^t \Phi^{-1}(\tau) \mathbf{b}(\tau) \, d\tau, \]
where \(\mathbf{x}_0\) is the solution to the homogeneous system.
3. **Matrix Exponential Relation:**
- In the case where \(\mathbf{A}(t)\) is constant (i.e., \(\mathbf{A}(t) = \mathbf{A}\)), the fundamental matrix solution is given by:
\[ \Phi(t) = e^{\mathbf{A}t}, \]
where \(e^{\mathbf{A}t}\) is the matrix exponential of \(\mathbf{A}t\).
In summary, the fundamental matrix \(\Phi(t)\) is a matrix solution to the linear differential system with initial condition \(\Phi(0) = \mathbf{I}\). It plays a crucial role in analyzing and solving systems of linear differential equations, especially when dealing with initial value problems and non-homogeneous terms.