EigenCalc

Eigenvalue Calculator

Enter a 2×2 matrix — get its eigenvalues from the characteristic equation.

2x2 matrix eigenvalues

a₁₁

a₁₂

a₂₁

a₂₂

Eigenvalue 1

Eigenvalue 2

Trace: 7
Determinant: 10

What you entered

Matrix: [[4, 1], [2, 3]]

1
Trace: tr(A) = a + d
4 + 3= 7
2
Determinant: det(A) = ad − bc
4×3 − 1×2= 10
3
Discriminant: tr² − 4·det
7² − 4×10= 9
4
λ₁ = (tr + √discriminant) / 2
(7 + √9) / 2= 5
5
λ₂ = (tr − √discriminant) / 2
(7 − √9) / 2= 2

Result

Eigenvalues: 5, 2

The eigenvalues are 5 and 2 — these are the scalars λ for which Av = λv has a non-trivial solution. The trace (sum of eigenvalues) is 7 and the determinant (product of eigenvalues) is 10.

Eigenvalues of a 2x2 matrix

Eigenvalues solve lambda squared minus trace times lambda plus determinant equals zero. Their sum is the trace and their product the determinant.

Matrix	Trace	Determinant	Eigenvalue 1	Eigenvalue 2
[[4,1],[2,3]]	7	10	5.0000	2.0000
[[2,0],[0,3]]	5	6	3.0000	2.0000
[[1,2],[2,1]]	2	-3	3.0000	-1.0000
[[5,4],[1,2]]	7	6	6.0000	1.0000
[[3,1],[0,3]]	6	9	3.0000	3.0000

Every row satisfies two checks worth using: the eigenvalues sum to the trace and multiply to the determinant. Row one gives 5 plus 2 equals 7 and 5 times 2 equals 10, both matching. Row two is diagonal, so its eigenvalues are just the diagonal entries. Row five has a repeated eigenvalue of 3, which happens when the discriminant is zero and signals a defective matrix with only one independent eigenvector. Eigenvalues are the directions a transformation merely stretches without rotating, which is why they underpin PCA, vibration analysis and stability.

What eigenvalues tell you

Eigenvalues λ of a matrix A satisfy Av = λv — they're the factors by which the matrix stretches its eigenvectors. For a 2×2 matrix, λ = (tr ± √(tr² − 4det)) / 2, where tr = a+d is the trace and det = ad−bc. If the discriminant is negative, eigenvalues are complex (rotation without a real stretch direction).

Why eigenvalues matter

Eigenvalues determine matrix behavior: all |λ| < 1 means the system decays; any |λ| > 1 means it grows. In PCA, eigenvalues of the covariance matrix rank dimensions by variance. In structural engineering, eigenvalues give natural vibration frequencies. Google's PageRank is the dominant eigenvector of the web graph's transition matrix.

Frequently asked questions

I have the matrix [[3, 1], [0, 2]] — what are its eigenvalues?

For a triangular matrix (zeros below the diagonal), eigenvalues are simply the diagonal entries: λ₁ = 3 and λ₂ = 2. In general, solve det(A − λI) = 0, which gives the characteristic equation λ² − (trace)λ + det = 0. Here: λ² − 5λ + 6 = 0, giving λ = 3 and λ = 2.

What do complex eigenvalues mean?

Complex eigenvalues (like 2 ± 3i) indicate the transformation involves rotation, not just stretching. A matrix with only real eigenvalues stretches along its eigenvectors; complex eigenvalues mean it spirals. The modulus |λ| determines whether the spiral grows (|λ| > 1) or shrinks (|λ| < 1).

How are eigenvalues related to the determinant and trace?

For a 2×2 matrix: the trace (sum of diagonal) equals the sum of eigenvalues, and the determinant equals their product. So det = λ₁ × λ₂ and trace = λ₁ + λ₂. If you know these two numbers, you can find eigenvalues via the quadratic formula without computing the full characteristic equation. The matrix determinant calculator finds one of these values.

Why does Google's PageRank use eigenvalues?

The web is modeled as a matrix where each entry represents the probability of following a link from one page to another. The dominant eigenvector (the one with the largest eigenvalue) gives each page's steady-state probability of being visited — that probability is the PageRank score.