What Eigenvalues and Eigenvectors Are
Most vectors change direction when a matrix acts on them. An eigenvector is special: the matrix only stretches or flips it, so A v = λ v. The stretch factor λ is the eigenvalue. The word comes from the German eigen, meaning “own” or “characteristic”, and was popularised by David Hilbert around 1904.
Think of pushing a swing: there is one rhythm (frequency) at which it responds most strongly. That frequency is an eigenvalue of the system.
How to Find Them
- Subtract λ from the diagonal and set det(A − λI) = 0. This gives the characteristic polynomial.
- Solve for λ.
- For each λ, solve (A − λI)v = 0 to get the eigenvectors.
A = [[4, 1], [2, 3]]
- Characteristic polynomial: λ² − 7λ + 10 = 0 (trace 7, determinant 10).
- Factor: (λ − 2)(λ − 5) = 0, so λ = 2 and λ = 5.
- For λ = 2: A − 2I = [[2, 1], [2, 1]], so 2x + y = 0 and v = (1, −2).
- For λ = 5: A − 5I = [[−1, 1], [2, −2]], so x = y and v = (1, 1).
Answer: λ = 2 with v = (1, −2); λ = 5 with v = (1, 1)
Two quick checks: the eigenvalues add up to the trace (2 + 5 = 7) and multiply to the determinant (2 × 5 = 10). The calculator performs both.
Irrational and Complex Eigenvalues
Not every matrix has neat eigenvalues. The Fibonacci matrix [[1, 1], [1, 0]] has characteristic polynomial λ² − λ − 1, so λ = (1 ± √5)/2, which are the golden ratio 1.618 and −0.618. That is why Fibonacci numbers grow by a factor of 1.618 each step.
A rotation matrix such as [[0, −1], [1, 0]] has λ = ±i. No real vector keeps its direction under a 90° turn, so the eigenvalues must be complex.
Diagonalization and Powers
If A has n independent eigenvectors, write A = P D P⁻¹, where P holds the eigenvectors as columns and D holds the eigenvalues on its diagonal. Powers then become easy: Aⁿ = P Dⁿ P⁻¹, and Dⁿ just raises each diagonal entry to the nth power.
D = diag(2, 5)
- D⁵ = diag(32, 3125).
- A⁵ = P D⁵ P⁻¹.
Answer: A⁵ = [[2094, 1031], [2062, 1063]]
A repeated eigenvalue is not always a problem, but [[2, 1], [0, 2]] has only one eigenvector direction. It is called defective and cannot be diagonalized.
Where Eigenvalues Are Used
- Google PageRank: the ranking of pages is the eigenvector for eigenvalue 1 of a link matrix.
- Markov chains: for the matrix [[0.9, 0.5], [0.1, 0.5]] the eigenvalues are 1 and 0.4. The eigenvector for 1 gives the long-run split of 5/6 and 1/6.
- Principal component analysis: eigenvectors of a covariance matrix are the main directions of variation in data.
- Vibrations and stability: eigenvalues decide whether a bridge, circuit or control system settles down or blows up.
What We Think Is Overrated, Underrated and Wrong
Worked Examples
For [[2, 1], [1, 2]] the characteristic polynomial is (2 − λ)² − 1 = 0, so λ = 3 and 1 with eigenvectors (1, 1) and (1, −1). Check: the trace 4 equals 3 + 1 and the determinant 3 equals 3 × 1.
The Fibonacci matrix [[1, 1], [1, 0]] has eigenvalues (1 ± √5) ÷ 2, and the larger one is the golden ratio 1.618. A rotation matrix has complex eigenvalues ±i, because no real direction stays fixed under a rotation.
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: hand-solving cubic characteristic polynomials. The trace and determinant checks matter more than the algebra.
Underrated: the trace and determinant check. The eigenvalues must sum to the trace and multiply to the determinant.
Plain wrong: the idea that every real matrix has real eigenvalues. Rotations do not.
Eigenvalue Mistakes That Cost Marks
- Subtracting λ from every entry instead of only the diagonal.
- Taking v = 0 as an eigenvector. Eigenvectors must be nonzero.
- Mismatching eigenvalues and eigenvectors when building P and D.
- Forgetting that eigenvectors can be scaled: (1, −2) and (2, −4) are the same direction.
- Expecting real eigenvalues from a rotation.
Why Students Get Stuck on Eigenvalues and Eigenvectors, and What a Good Calculator Should Do
Searching for help with eigenvalues and eigenvectors usually starts late at night with one problem that will not work out. The characteristic polynomial is easy to set up and easy to mis-solve. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to find eigenvalues, then eigenvectors, and know what they mean, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Forgetting to subtract λ only on the diagonalEigenvalues of [[2,1],[1,2]]↗
- Solving the characteristic polynomial incorrectlyEigenvalues of [[4,1],[2,3]]↗
- Not knowing how to turn an eigenvalue into an eigenvectorEigenvalues of [[1,2],[3,4]]↗
- Not seeing what an eigenvector does geometricallyEigenvalues of [[3,0],[0,5]]↗
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants the characteristic polynomial written out
- Wants each eigenvector found with its scaling
- Wants a quick check by multiplying back
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Eigenvalues of [[2,1],[1,2]]↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.