Learning objective: Find eigenvalues and eigenvectors, interpret the characteristic determinant, and check your answer without relying on memorised roots.
Understand the idea
An eigenvector identifies a direction that a linear transformation stretches without rotating away from that direction. Eigenvalues quantify the stretching. In circuit state-space models and control-system stability analysis, eigenvalues also indicate how individual modes evolve with time. For a square matrix A, we seek nonzero v such that Av = λv.
Worked example: given data
- Consider A = [[4, 1], [2, 3]].
- Determine both eigenvalues and one eigenvector for each.
- Check the answers by evaluating Av and λv.
Detailed solution
Form the characteristic equation
Set det(A − λI) = 0. In this case, (4 − λ)(3 − λ) − (1)(2) = 0. Expanding gives λ² − 7λ + 10 = 0. The trace 7 and determinant 10 provide a quick check on the coefficients.
Solve the roots
Factor λ² − 7λ + 10 = (λ − 5)(λ − 2). The eigenvalues are λ = 5 and λ = 2. Their sum is the trace, and their product is the determinant, as required.
Find the eigenvector for λ = 5
Substitute into (A − 5I)v = 0 to obtain −x + y = 0. Choose x = 1, y = 1. Thus v₅ = [1, 1]ᵀ. Any nonzero multiple is also an eigenvector.
Find the eigenvector for λ = 2
Now (A − 2I)v = 0 yields 2x + y = 0. Choose x = 1, y = −2, so v₂ = [1, −2]ᵀ.
Verify both answers
A[1,1]ᵀ = [5,5]ᵀ = 5[1,1]ᵀ. Also A[1,−2]ᵀ = [2,−4]ᵀ = 2[1,−2]ᵀ. Both checks match their defining equations.
What the answer means
A pair of distinct eigenvalues produces two linearly independent eigenvectors in this example. For a continuous-time state model, the signs of the real parts of eigenvalues influence whether modes decay or grow; however, that stability interpretation applies to a system matrix in a differential equation, not arbitrarily to every matrix appearing in a calculation.
Three mistakes to avoid
- Subtracting λ from every matrix entry rather than only from the main diagonal.
- Confusing an eigenvector with a unique vector; multiplying it by any nonzero scalar gives another.
- Reporting roots without checking the trace and determinant.
Try a related question
Try A = [[3, 0], [0, −1]]. What are its eigenvalues and convenient eigenvectors?
Show the worked-answer check
The eigenvalues are 3 and −1. Corresponding eigenvectors are [1, 0]ᵀ and [0, 1]ᵀ. A diagonal matrix makes the verification immediate.
How to revise this concept
Review matrix determinants and characteristic polynomials, then practise state-space eigenvalue questions. First attempt the example on paper without opening the answer, then compare the intermediate steps. Explain each assumption before memorising a formula.
Content note: This is a newly prepared practice example, not an official GATE previous-year question. It was drafted with AI assistance for this educational website. Students should verify calculations and exam requirements against standard textbooks and official sources; educator review is recommended before publication.