When you substitute y = e^{λt}v into the system y' = Ay, both sides reduce to the same equation. What is that equation, and why does it make the eigenvalue method work?
AIt reduces to Av = v, meaning only unit eigenvectors are valid — this constrains which exponentials solve the system
BIt reduces to Av = λv, the eigenvalue equation — the system has a solution of this form exactly when v is an eigenvector of A with eigenvalue λ
CIt reduces to A = λI, meaning the method only works when A is a scalar multiple of the identity
DIt reduces to λv = 0, so only the trivial solution exists unless λ = 0
Substituting y = e^{λt}v gives y' = λe^{λt}v and Ay = Ae^{λt}v = e^{λt}Av. Setting these equal and dividing by the nonzero scalar e^{λt} gives Av = λv — the eigenvalue equation. This is why the method works: the exponential structure perfectly matches the system's structure. The scalar λ absorbs the time derivative, and the eigenvector condition ensures the matrix product Av produces the same vector back (scaled), so the equation is self-consistent.
Question 2 Multiple Choice
The matrix A has a complex eigenvalue λ = 2 + 3i with eigenvector v = p + qi (p, q real vectors). How do you obtain real-valued solutions from this?
AUse only the real part of e^{λt}v and discard the imaginary part
BReplace λ with its real part 2 and ignore the imaginary part 3i entirely
CApply Euler's formula to e^{(2+3i)t}(p+qi) and take its real and imaginary parts as two independent real-valued solutions
DComplex eigenvalues indicate the system has no real-valued solutions, so a different method is needed
Euler's formula gives e^{(2+3i)t}(p+qi) = e^{2t}[(cos 3t)p − (sin 3t)q] + ie^{2t}[(sin 3t)p + (cos 3t)q]. The real and imaginary parts are each real-valued solutions, and together they form two linearly independent solutions replacing the complex conjugate pair. This is why complex eigenvalues produce oscillatory behavior — the sin and cos terms encode rotation in the solution space. Option D is wrong: complex eigenvalues always occur in conjugate pairs for real A, and extracting real/imaginary parts always yields real solutions.
Question 3 True / False
If the matrix A has 3 distinct real eigenvalues, the general solution to y' = Ay is a linear combination of 3 independent exponential-vector solutions e^{λ₁t}v₁, e^{λ₂t}v₂, e^{λ₃t}v₃.
TTrue
FFalse
Answer: True
Distinct eigenvalues guarantee linearly independent eigenvectors, so the 3 solutions e^{λᵢt}vᵢ are independent and span the full 3-dimensional solution space. Every solution is a unique linear combination c₁e^{λ₁t}v₁ + c₂e^{λ₂t}v₂ + c₃e^{λ₃t}v₃, with the constants determined by initial conditions. This is the diagonalization of the system: in the eigenvector basis, the three equations decouple into independent scalar ODEs, each solved by its own exponential.
Question 4 True / False
Complex eigenvalues of A mean the system y' = Ay has no real-valued solutions, and the eigenvalue method can seldom be applied.
TTrue
FFalse
Answer: False
Complex eigenvalues do not prevent real solutions — they guarantee oscillatory ones. When λ = α + βi is complex, the complex solution e^{λt}v is separated into real and imaginary parts using Euler's formula, yielding two real-valued solutions involving e^{αt}cos(βt) and e^{αt}sin(βt). The eigenvalue method applies exactly as before; the extra step is extracting these real-valued solutions from the complex result. Complex eigenvalues are in fact the typical case for systems with oscillatory behavior like springs, circuits, and predator-prey models.
Question 5 Short Answer
Explain why the eigenvalue method effectively 'decouples' a system of coupled ODEs. What does working in the eigenvector basis reveal about the structure of the solutions?
Think about your answer, then reveal below.
Model answer: In the eigenvector basis, the matrix A acts by simply scaling each coordinate by its corresponding eigenvalue. This means the n coupled equations separate into n independent scalar equations, each of the form z'ᵢ = λᵢzᵢ, solved independently by zᵢ = cᵢe^{λᵢt}. The coupling in the original coordinates disappears because eigenvectors are precisely the directions A doesn't rotate — only scales. The general solution is then a superposition of these decoupled exponentials, transformed back into original coordinates using the eigenvectors.
The coupling in y' = Ay comes from off-diagonal entries of A — the rate of change of one variable depends on others. Eigenvectors are the special directions where this cross-dependence vanishes: Avᵢ = λᵢvᵢ means A maps vᵢ back to itself (scaled), with no mixing into other directions. Expressing the solution as a combination of these directions diagonalizes A, separating the system into independent 1D problems. This is the deep connection between the eigenvalue method and matrix diagonalization.