How to use the Eigenvalue and Eigenvector Calculator
- 1. Enter the matrixType one row per line, with numbers separated by spaces. Fractions like 1/2 work.
- 2. Press Find eigenvaluesThe tool builds |A − λI| = 0 and solves it, including repeated and complex roots.
- 3. Read the eigenvectorsEach real eigenvalue gets its eigenvector(s), scaled so the smallest entry is 1.
How eigenvalues are found
An eigenvalue λ of a square matrix A is a number for which A·X = λX has a non-zero solution X. That happens only when |A − λI| = 0, the characteristic equation. For a 2×2 matrix it is λ² − (trace)λ + |A| = 0. For a 3×3 matrix it is λ³ − (trace)λ² + (sum of principal 2×2 minors)λ − |A| = 0.
Each eigenvector is a non-zero solution of (A − λI)X = 0. The tool finds it by row reducing A − λI, so repeated eigenvalues show every independent eigenvector.
Worked example
Take the default matrix A = [8 −6 2; −6 7 −4; 2 −4 3]. The trace is 8 + 7 + 3 = 18. The principal minors are (7·3 − 16) = 5, (8·3 − 4) = 20 and (8·7 − 36) = 20, which add up to 45. The determinant is 0. So the characteristic equation is λ³ − 18λ² + 45λ = 0, or λ(λ − 3)(λ − 15) = 0.
The eigenvalues are 0, 3 and 15. For λ = 0 the eigenvector is (1, 2, 2), for λ = 3 it is (2, 1, −2), and for λ = 15 it is (2, −2, 1). As a check, 0 + 3 + 15 = 18 = trace and 0 × 3 × 15 = 0 = |A|.
Repeated and complex eigenvalues
If λ repeats, the matrix may have one eigenvector or two. The table shows both counts: algebraic multiplicity and independent eigenvectors. When there are fewer eigenvectors than the multiplicity, the matrix cannot be diagonalised. A real matrix can also have complex eigenvalues, such as ±i for a 90° rotation; the tool lists them as a ± bi. To multiply or invert matrices, use the matrix multiplication visualizer or the matrix rank calculator.
Frequently asked questions
Why is my eigenvector different from the textbook?
Any non-zero multiple of an eigenvector is also an eigenvector. The tool scales so the smallest non-zero entry is 1, so (2, 1, −2) and (−4, −2, 4) are the same answer.
Can it do 4×4 matrices?
No. It solves 2×2 and 3×3 matrices exactly through the characteristic polynomial, which covers university engineering maths exams.
What does a zero eigenvalue mean?
The matrix is singular: its determinant is 0 and it has no inverse.
Is this calculator free?
Yes. You enter your name and WhatsApp number once when you see your result, and other Uncodemy tools then open without the form.