State Space Equations
State space equations are mathematical models used to describe the behavior of a dynamic system. They consist of a set of first-order differential equations that relate the system's inputs, outputs, and states.

The general form of state space equations is:
dx/dt = Fx + Gu
y = Hx + Ju

Where:
- x is the state vector, representing the internal variables of the system.
- u is the input vector, representing the system's inputs.
- y is the output vector, representing the system's outputs.
- F is the system matrix, relating the states to their derivatives.
- G is the input matrix, relating the inputs to the state derivatives.
- H is the output matrix, relating the states to the outputs.
- J is the direct transmission matrix, relating the inputs to the outputs.

Closed Form Solution
A closed form solution of the state space equations means finding an analytical expression for the state variables as a function of time. In other words, it provides a direct formula to calculate the state variables at any given time without the need for numerical integration.

Properties of the System Matrix F
The system matrix F plays a crucial role in determining the nature of the state space equations and whether a closed form solution is easily obtained or not. Let's examine the properties of the system matrix:

1. Transpose Matrix (Option A)
The transpose of a matrix is obtained by interchanging its rows with columns. Transposing the system matrix F does not necessarily lead to a closed form solution. It may introduce complex dependencies between the state variables, making it difficult to find an explicit expression for the states.

2. Symmetric Matrix (Option B)
A symmetric matrix is equal to its own transpose. While symmetric matrices have certain desirable properties, such as real eigenvalues and orthogonal eigenvectors, they do not guarantee a closed form solution. The system matrix F being symmetric does not simplify the solution process.

3. Identity Matrix (Option C)
The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. If the system matrix F is an identity matrix, the state space equations simplify to dx/dt = x + Gu, and y = Hx + Ju. In this case, the closed form solution is easily obtained by integrating the first equation.

4. Diagonal Matrix (Option D)
A diagonal matrix is a square matrix with zeros off the diagonal and arbitrary values on the diagonal. When the system matrix F is diagonal, the state space equations decouple, meaning that the state variables evolve independently of each other. This allows for a simple closed form solution by integrating each equation separately.

Conclusion
Among the given options, the system matrix F being diagonal guarantees a closed form solution of the state space equations. The diagonal matrix property simplifies the equations and allows for independent integration of each state variable. Other options, such as transpose, symmetric, or identity matrix, do not necessarily lead to a closed form solution and may introduce additional complexities in finding the explicit expression for the states.