Questions: State Transition Matrix

Question 1

A linear system has state matrix A with eigenvalues λ₁ = −3 and λ₂ = 1. What does the state transition matrix e^{At} imply about the system's free response?

• A. Both modes decay, so the system is stable for any initial condition
• B. One mode decays (λ₁ = −3) and one grows unboundedly (λ₂ = 1), making the system unstable
• C. The system oscillates because the eigenvalues have opposite signs
• D. The eigenvalues determine only the input response, not the free response

Answer: (B)

The free response x(t) = e^{At}x(0) decomposes into modes e^{λᵢt}. A negative real eigenvalue (λ₁ = −3) produces a decaying mode e^{−3t}→0; a positive real eigenvalue (λ₂ = 1) produces a growing mode e^{t}→∞. Any initial condition with a nonzero component in the unstable eigendirection causes the state to diverge. One unstable mode is sufficient to make the entire system unstable.

Question 2

How is e^{At} computed when A is diagonalizable as A = PΛP⁻¹?

• A. Apply the scalar exponential to each entry of A to get e^{At}
• B. Compute e^{At} = Pe^{Λt}P⁻¹, where e^{Λt} is the diagonal matrix with entries e^{λᵢt}
• C. Compute e^{At} = e^P · Λt · e^{−P} using separate matrix exponentials of P
• D. Multiply each entry of At by e and sum the result

Answer: (B)

For diagonalizable A = PΛP⁻¹, the matrix exponential factors as e^{At} = Pe^{Λt}P⁻¹. Since Λ is diagonal, e^{Λt} is simply the diagonal matrix with scalar exponentials e^{λᵢt} on the diagonal — straightforward to compute. The columns of P are eigenvectors defining the natural mode directions, and the diagonal entries of e^{Λt} give each mode's time evolution.

Question 3

The state transition matrix Φ(t) = e^{At} satisfies Φ(0) = I, the identity matrix.

• T. True
• F. False

Answer: True

At t = 0, e^{A·0} = e^0 = I by the matrix exponential definition (I + 0 + 0 + ... = I). Physically, this means at t = 0, no evolution has occurred — the state equals the initial state x(0). This is directly analogous to the scalar identity e^0 = 1, and is one of the key defining properties of the state transition matrix.

Question 4

The matrix exponential e^{At} can be computed by applying the scalar function eˣ to each element of the matrix At individually.

• T. True
• F. False

Answer: False

This is the most common misconception about the matrix exponential. e^{At} is defined through the matrix Taylor series: I + At + (At)²/2! + (At)³/3! + ... The matrix powers (At)ⁿ involve matrix multiplication, coupling all elements together. Element-wise exponentiation gives completely wrong results except for diagonal matrices. The fact that diagonal A happens to match element-wise exponentiation is why the misconception persists, but it fails for any non-diagonal A.

Question 5

Explain why the eigenvalues of A — not the individual matrix entries — determine the long-term behavior of the system's free response x(t) = e^{At}x(0).

Model answer: When A is diagonalizable, the eigenvectors of A form a natural coordinate basis for the state space. In this basis, each mode evolves independently as e^{λᵢt}, where λᵢ is the corresponding eigenvalue. The free response decomposes into a sum of modal contributions, each growing or decaying at a rate determined by the real part of its eigenvalue. A negative real part gives decay, positive gives growth, purely imaginary gives oscillation. The individual matrix entries contain this eigenvalue information implicitly — but a similarity transformation A → PAP⁻¹ can change all entries while leaving eigenvalues and system behavior unchanged.

This is why system analysis centers on eigenvalues. They are the coordinate-independent, physically meaningful quantities: changing the basis (coordinates) of the state space changes the matrix entries but cannot change whether modes grow or decay. The eigenvalues are what the physics depends on; the matrix entries depend on the chosen coordinate system.