Eigenvector Calculator

Square matrix. Rows separated by a semicolon or newline, entries by spaces or commas.
Eigenvectors
5: (1, 1) | 2: (1, -2)
Eigenvalue and Eigenvector Calculator
Eigenvalues
5, 2
Characteristic polynomial
λ^2 - 7λ + 10
Order (n)
2
Trace (sum of eigenvalues)
7
Determinant (product of eigenvalues)
10
Spectral radius (largest |lambda|)
5
Largest real eigenvalue
5
Smallest real eigenvalue
2
Symmetric
no
Has complex eigenvalues
no
Diagonalisable
yes
Algebraic / geometric multiplicities
5: algebraic 1, geometric 1; 2: algebraic 1, geometric 1
D (eigenvalues on the diagonal)
5 0; 0 2
P (eigenvectors as columns)
1 1; 1 -2
Worked out from Gilbert Strang, Introduction to Linear Algebra, ch. 6 (eigenvalues and eigenvectors) and 2 other sources.

How this is worked out

10 steps
  1. Matrix is 2x2.
  2. [ 4 1 ]
  3. [ 2 3 ]
  4. Characteristic polynomial det(λI - A) = λ^2 - 7λ + 10 = 0.
  5. Roots of the characteristic polynomial give the eigenvalues.
  6. Each eigenvector spans the null space of A - λI, found by Gauss-Jordan elimination in the complex plane.
  7. λ = 5 (algebraic multiplicity 1, geometric multiplicity 1): (1, 1)
  8. λ = 2 (algebraic multiplicity 1, geometric multiplicity 1): (1, -2)
  9. A = P D P⁻¹ with D = 5 0; 0 2 and P = 1 1; 1 -2.
  10. Check: the eigenvalues sum to the trace (7) and multiply to the determinant (10).
Eigenvalues (raw)table
realimaginarymodulustext
5055
2022

Results are general information, not financial, tax, legal, medical or engineering advice.