Build 2×2 matrices and read their eigenvalues, eigenvectors, trace and determinant in the explorer, and predict — before you read it off the screen — the eigenvalues of a symmetric matrix, whether its eigenvectors are orthogonal, that the trace equals the sum and the determinant the product of the eigenvalues, and why a rotation has no real eigen-direction.
An eigenvector of A is a direction the matrix only stretches, never rotates: A v = λ v, with the eigenvalue λ the stretch factor. The eigenvalues are the roots of the characteristic polynomial λ² - tr(A) λ + det(A) = 0, so its discriminant tr(A)² - 4 det(A) decides the kind: positive gives two real eigenvalues, negative a complex pair — a rotation, with no real direction left fixed. Two identities always hold: tr(A) = λ₁ + λ₂ and det(A) = λ₁ λ₂. And a symmetric matrix always has real eigenvalues with orthogonal eigenvectors.
Open the explorer. Each step names the 2×2 matrix A to set from its four entries a b c d; type them into the matrix inputs, and read the eigenvalue, eigenvector, trace and determinant readouts.
Matrix A -> [ a b ; c d ] (set the four entries each step names)
Read: eigenvalues, eigenvectors, tr(A), det(A), discriminant
Watch the eigenvalue, eigenvector, trace, determinant and discriminant readouts in the main panel — every number you predict is printed there.
Set A = [[2, 1], [1, 2]] — symmetric. Its eigenvalues solve det(A - λ I) = 0, i.e. λ² - 4λ + 3 = 0. Predict the larger eigenvalue λ₁, then read the readout.
The readout shows the pair {3, 1}, so λ₁ = 3: the polynomial factors as (λ - 3)(λ - 1). Both roots are real — a symmetric matrix always has real eigenvalues.
Keep A = [[2, 1], [1, 2]]. Read its two eigenvectors — one along (1, 1) (for λ = 3), one along (1, -1) (for λ = 1). Predict their dot product, then read the readouts.
The dot product is (1)(1) + (1)(-1) = 0, so the two eigenvectors are orthogonal. Eigenvectors of a symmetric matrix (for distinct eigenvalues) are always perpendicular.
Set A = [[3, 1], [2, 2]]. Its trace is tr(A) = 3 + 2 = 5. Predict λ₁ + λ₂, then read the eigenvalues.
The eigenvalues are {4, 1}, and their sum is 4 + 1 = 5 — exactly tr(A). This identity tr(A) = λ₁ + λ₂ holds for every matrix.
Keep A = [[3, 1], [2, 2]]. Its determinant is det(A) = 3·2 - 1·2 = 4. Predict λ₁ λ₂, then read it.
The product is 4·1 = 4 — exactly det(A). This identity det(A) = λ₁ λ₂ holds for every matrix, so det(A) = 0 means one eigenvalue is 0.
Set A = [[0, -1], [1, 0]] — a 90° rotation. Its discriminant is tr(A)² - 4 det(A) = 0 - 4. Predict the discriminant, then read the readout and the case verdict.
The discriminant is 0 - 4 = -4, which is negative, so the eigenvalues are the complex pair ±i and there is no real eigenvector — a rotation turns every direction, leaving none fixed.
Everything above is waiting in the explorer. Enter any 2×2 matrix, watch the eigen-directions and the image of the unit circle update in real time, and see the eigenvalues, eigenvectors, trace, determinant and the real/complex/repeated verdict change at once — then start the rotation animation to watch a vector chase its eigen-direction.
- Found the real eigenvalues {3, 1} of the symmetric [[2, 1], [1, 2]] — top eigenvalue 3
- Saw its eigenvectors are orthogonal — dot product 0
- Confirmed tr(A) = λ₁ + λ₂ = 5 on [[3, 1], [2, 2]]
- Confirmed det(A) = λ₁ λ₂ = 4 on the same matrix
- Saw a rotation's negative discriminant -4 — a complex pair, no real eigen-direction
- Every value you predicted is the demo's own eigen-arithmetic, not a picture