Eigenvalues & Eigenvectors

The last lesson ended on a hint: most directions a matrix shoves off to a new angle, but a few it leaves sitting on their own line. Those special directions are eigenvectors, and the factor by which each one is stretched is its eigenvalue λ. Push a random arrow through a matrix and it comes out rotated and stretched; push an eigenvector through and it comes out pointing the very same way, only longer or shorter. That single idea powers covariance, PCA, PageRank, and the stability of dynamical systems — and it shows up almost every year in GATE DA.

The defining equation

A non-zero vector v is an eigenvector of A when applying A only scales it:

A v = λ v        (v ≠ 0)

The scalar λ says by how much: positive keeps the direction, negative flips it, a small λ shrinks it. We insist v ≠ 0 because A·0 = λ·0 holds for every λ and so tells us nothing.

Play with it below. The fan of faint arrows shows where the matrix sends every input direction — most swing to a new angle, but the highlighted lines stay put and only scale by their eigenvalue. Drag the test vector onto one to feel a direction survive; switch to the rotation preset and the eigenlines vanish — those eigenvalues are complex, more on that shortly.

Finding them — the characteristic polynomial

Rearrange A v = λ v into (A − λI) v = 0. A non-zero v can solve this only if A − λI is singular — and singular, from the determinant lesson, means

det(A − λI) = 0

This is the characteristic polynomial. For a 2×2 matrix it is always a quadratic, and it has a form worth memorising on sight:

λ² − trace(A)·λ + det(A) = 0

Once you have the eigenvalues, plug each back into (A − λI)v = 0 and solve the linear system for the eigenvector direction v. Two shortcuts ride along, and they earn a large share of the marks on this topic — for any n×n matrix with eigenvalues λ₁, …, λₙ:

So trace(A) = Σλᵢ and det(A) = Πλᵢ. They let you check an answer instantly, and often find the second eigenvalue the moment you know the first.

A worked example — a real GATE DA 2024 question

Find the eigenvalues of M = [[2, −1], [3, 1]].

Read off the trace and determinant, build the quadratic, check the discriminant:

trace(M) = 2 + 1 = 3
det(M)   = (2)(1) − (−1)(3) = 2 + 3 = 5

λ² − 3λ + 5 = 0      discriminant = (−3)² − 4·5 = 9 − 20 = −11 < 0

λ = (3 ± √(−11)) / 2 = (3 ± i√11)/2     → a complex conjugate pair

The negative discriminant means no real eigenvalues — a complex conjugate pair (this is GATE DA 2024). Geometrically M rotates as well as scales, so no real direction survives, exactly the rotation preset in the widget. For contrast, a clean real case — a triangular matrix gives its eigenvalues for free:

A = [[2, 1],     det(A − λI) = (2−λ)(3−λ) − 0 = 0   →  λ = 2 or λ = 3
     [0, 3]]

For any triangular matrix (zeros below or above the diagonal) the eigenvalues are just the diagonal entries — no algebra. Check: trace = 5 = λ₁+λ₂ ✓, det = 6 = λ₁·λ₂ ✓.

A question to carry forward

Eigenvalues are not just a number to find — they are a handle on the matrix itself. Here is the thread onward: if you know the eigenvalues of A, what are the eigenvalues of A², of A⁻¹, of A + 5I? And what makes a symmetric matrix so much better behaved than a general one that its eigenvectors come out perpendicular?

In one breath

• Eigenvector v ≠ 0: a direction the matrix only stretches, A v = λ v; the eigenvalue λ is the stretch factor.
• Find via det(A − λI) = 0; for a 2×2, λ² − trace·λ + det = 0. Then solve (A−λI)v = 0 for the direction.
• Shortcuts: trace = Σλ (sum), det = Πλ (product) — check answers, or get the second eigenvalue from the first.
• Triangular matrix → eigenvalues are the diagonal entries.
• Traps: a real matrix can have a complex conjugate pair (trace²−4det < 0); eigenvectors are non-unique (any non-zero scalar multiple).

Practice