Question

Which of the following matrices is the exponential of a Q-matrix? (a) ( 1 0 ) (0 1 ) (b) ( 1 0 ) ( 1 0) (c) ( 0 1 ) ( 1 0 )

Answer 1

The identity matrix (a) is the exponential of the zero matrix, a special case of a Q-matrix, because it upholds the necessary properties: non-negativity and row sums equal to one. None of the other matrices presented have these properties in full.

Step-by-step explanation:

The exponential of a Q-matrix, typically found in the study of Markov processes or related fields in mathematics, must meet certain criteria. A Q-matrix, also known as a rate matrix or generator matrix, is a square matrix where the off-diagonal entries are non-negative and each row sums to zero. Matrices that can be considered exponentials of Q-matrices would display a transition of probabilities with certain properties, such as non-negativity and row sums equal to one since they represent probability distributions.

Matrices presented are:

• (a)
  \( \left(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right) \) - An identity matrix.
• (b)
  \( \left(\begin{matrix} 1 & 0 \\ 1 & 0 \end{matrix}\right) \) - Not a valid exponential of a Q-matrix because the rows do not sum to one.
• (c)
  \( \left(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}\right) \) - This matrix has rows that sum to one but is not non-negative.

The correct answer would be matrix (a) since it is the identity matrix, which is indeed the exponential of the zero matrix, a special case of a Q-matrix with all entries equal to zero. The identity matrix is the only matrix among the options that can be the result of exponentiating a Q-matrix, since it represents no change in the state of a probabilistic system over time (i.e., a probability of 1 that the system remains in its current state).

Answer 2

Option (a) is the correct answer.

A Q-matrix, or infinitesimal generator matrix, is a square matrix whose off-diagonal elements are non-negative and each row sums to zero.

Let's evaluate each of the given matrices:

(a)
`\( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)`

Already a valid transition probability matrix as it's an identity matrix. This matrix is already diagonal, and its off-diagonal entries are zero. Therefore, it is a Q-matrix, and its exponential is
`e^(Qt)` for any value of t.

(b)
`\( \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} \)`

This matrix does not have non-positive off-diagonal entries. Therefore, it is not a Q-matrix.

(c)
`\( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)`

This matrix has non-positive off-diagonal entries, but the sum of each row is not equal to zero. So, it is not a Q-matrix.

For a matrix to be the exponential of a Q-matrix, it must be a valid transition probability matrix. This means that all elements must be non-negative, and each row must sum to 1.

Therefore, the matrix
`\( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)` is the exponential of a Q-matrix.