Assertion (A): The eigen values of a linear continuous-data time invariant system controls the stability of the system.

Reason (R): The roots of the characteristic equation are the same as the eigen values of system matrix A of the state equations.

To determine whether the given assertion and reason are true or false, let's analyze each statement individually.

Statement A: The eigen values of a linear continuous-data time invariant system control the stability of the system.

Explanation:
In control theory, the stability of a system is a crucial property that determines whether the system will exhibit bounded or unbounded behavior over time. A system is considered stable if its response remains bounded for all bounded inputs.

Eigenvalues are a fundamental concept in linear algebra and are used to analyze the behavior of linear systems. In the context of control systems, the eigenvalues of the system matrix A play a crucial role in determining the stability of the system.

The eigenvalues of A can be obtained by solving the characteristic equation det(sI - A) = 0, where s is a complex variable and I is the identity matrix. The solutions to this equation are the eigenvalues of A.

For a linear continuous-data time invariant system, the eigenvalues of A can be classified into three categories based on their location in the complex plane:

1. Stable: If all the eigenvalues have negative real parts, then the system is stable. The system's response will decay over time, and any bounded input will result in a bounded output.

2. Unstable: If at least one eigenvalue has a positive real part, then the system is unstable. The system's response will grow exponentially over time, and even a bounded input can lead to an unbounded output.

3. Marginally stable: If the eigenvalues have zero real parts and are purely imaginary, the system is marginally stable. The response neither decays nor grows exponentially but oscillates indefinitely.

Therefore, the assertion that the eigenvalues of a linear continuous-data time invariant system control the stability of the system is true.

Statement R: The roots of the characteristic equation are the same as the eigenvalues of system matrix A of the state equations.

Explanation:
The characteristic equation of a linear continuous-data time invariant system is obtained by setting the determinant of the matrix (sI - A) equal to zero, where s is a complex variable and A is the system matrix. The roots of this equation are referred to as the characteristic roots or eigenvalues of the system.

The eigenvalues of the system matrix A are obtained by solving the characteristic equation. The eigenvalues represent the poles of the transfer function of the system, which describe its dynamic behavior.

Hence, the reason that the roots of the characteristic equation are the same as the eigenvalues of system matrix A of the state equations is true.

Conclusion:
Both the assertion and the reason are true, and the reason is a correct explanation of the assertion. The eigenvalues of a linear continuous-data time invariant system indeed control the stability of the system, and the roots of the characteristic equation correspond to the eigenvalues of the system matrix A.