acalculator

What are the eigenvectors?

Free online eigenvalue and eigenvector calculator. Fast, accurate, and easy to use math tool.

Your numbers

Matrix A
Eigenvalues
1, 3

The eigenvalues are 1, 3.

λ1
1
v1
[1, 0]
λ2
3
v2
[0.7071067812, 0.7071067812]
Characteristic polynomial
λ² - 4λ + 3
Trace
4
Determinant
3
Working
Characteristic polynomial: det(λI − A) = λ² - 4λ + 3; its roots are λ = 1, 3; for each λ, solve (A − λI)v = 0 and scale v to length 1

Eigenvalues: 1, 3. The eigenvalues are 1, 3.

How it is worked out

How to calculate

Finds every eigenvalue of a square matrix (2×2 to 6×6), real or complex, with a unit eigenvector for each, and the characteristic polynomial.

Example with the default inputs (Matrix A [1, 2; 0, 3]): The eigenvalues are 1, 3.

Method: Eigenvalues are the roots of det(λI − A) = 0; for each eigenvalue λ, an eigenvector is a nonzero v with (A − λI)v = 0, scaled to length 1.

  • The matrix is square, 2×2 to 6×6, with real entries. Complex eigenvalues come in conjugate pairs and are shown as a + bi, the one with the negative imaginary part first.
  • Eigenvalues are found numerically: the Jacobi method for symmetric matrices, the QR algorithm otherwise. A repeated eigenvalue that is an exact root of the characteristic polynomial (checked in fractions) is shown exactly; other k-fold eigenvalues of a defective matrix are accurate to about 16 ÷ k significant figures.
  • The characteristic polynomial and the determinant are exact: entries are read as fractions of their decimals.
  • Each eigenvector has length 1; its first component of largest size is made real and positive.
  • A repeated eigenvalue of a symmetric matrix gets orthonormal eigenvectors. For another matrix, an exactly found repeated eigenvalue gets a basis of its exact eigenspace, one vector per repeat while there are independent ones, then the last one again.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Matrix A 1, 2; 0, 3 gives Eigenvalues 1, 3, λ1 1, λ2 3, v1 [1, 0], v2 [0.7071067812, 0.7071067812], Characteristic polynomial λ² - 4λ + 3, Trace 4, Determinant 3.Source: Gilbert Strang, Introduction to Linear Algebra, 5th ed., §6.1 Introduction to Eigenvalues. https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_6-1.pdf
  2. Matrix A 1, 0, 0; 0, 2, 0; 0, 0, 3 gives Eigenvalues 1, 2, 3, v1 [1, 0, 0], v2 [0, 1, 0], v3 [0, 0, 1], Determinant 6.Source: Gilbert Strang, Introduction to Linear Algebra, 5th ed., §6.1 Introduction to Eigenvalues. https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_6-1.pdf
  3. Matrix A 4, 1; 2, 3 gives Eigenvalues 2, 5, v1 [-0.4472135955, 0.894427191], v2 [0.7071067812, 0.7071067812].Source: Gilbert Strang, Introduction to Linear Algebra, 5th ed., §6.1 Introduction to Eigenvalues. https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_6-1.pdf
  4. Matrix A 0, -1; 1, 0 gives Eigenvalues -i, i, λ1 -i, λ2 i, Characteristic polynomial λ² + 1, Determinant 1.Source: Gilbert Strang, Introduction to Linear Algebra, 5th ed., §6.1 Introduction to Eigenvalues. https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_6-1.pdf
  5. Matrix A 2, 1, 0; 1, 2, 1; 0, 1, 2 gives λ1 0.585786, λ2 2, λ3 3.414214, v2 [0.7071067812, 0, -0.7071067812], Characteristic polynomial λ³ - 6λ² + 10λ - 4.Source: Gilbert Strang, Introduction to Linear Algebra, 5th ed., §6.1 Introduction to Eigenvalues. https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_6-1.pdf
  6. Matrix A 1, 1; 0, 1 gives Eigenvalues 1, 1, v1 [1, 0], v2 [1, 0], Characteristic polynomial λ² - 2λ + 1, Note λ = 1 repeats twice but has only 1 independent eigenvector. So A is not diagonalizable, and the same eigenvector is shown again for the missing ones..Source: Gilbert Strang, Introduction to Linear Algebra, 5th ed., §6.1 Introduction to Eigenvalues. https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_6-1.pdf
  7. Matrix A 2, 0, 0, 0; 0, 3, 0, 0; 0, 0, 5, 0; 0, 0, 0, 7 gives Eigenvalues 2, 3, 5, 7, Characteristic polynomial λ⁴ - 17λ³ + 101λ² - 247λ + 210, Determinant 210, Trace 17.Source: Gilbert Strang, Introduction to Linear Algebra, 5th ed., §6.1 Introduction to Eigenvalues. https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_6-1.pdf

How it works

An eigenvalue λ of a square matrix A and an eigenvector v ≠ 0 satisfy Av = λv.

  1. Characteristic polynomial. The eigenvalues are the roots of p(λ) = det(λI − A). For an n×n matrix it has degree n, so there are n eigenvalues, counting repeats. For a 2×2 matrix, p(λ) = λ² − (trace)λ + (determinant).
  2. Eigenvalues. The calculator finds the roots numerically: the Jacobi method for a symmetric matrix, and the QR algorithm with shifts for any other matrix (Golub and Van Loan, Matrix Computations, sections 7.5 and 8.5). Complex eigenvalues of a real matrix come in conjugate pairs a ± bi.
  3. Eigenvectors. For each eigenvalue, it solves (A − λI)v = 0 and scales v to length 1. Any nonzero multiple of v is also an eigenvector, so the page fixes one: the first component with the largest size is made real and positive.

The page shows:

  • Eigenvalues in ascending order of real part, then imaginary part, with repeats (λ1, λ2, …). A real eigenvalue is a number, shown to 10 significant figures; a complex one is text such as 1 + 2i.
  • v1, v2, …: a unit eigenvector for each eigenvalue, in the same order.
  • Working: the polynomial, its roots, and how the eigenvectors are found, also listed step by step under "How it is worked out".
  • Characteristic polynomial det(λI − A), with the trace and determinant. The coefficients and the determinant are computed exactly: each entry is read as the fraction of its decimal (0.1 is 1/10) and the Faddeev-LeVerrier recurrence runs in fractions (M₁ = I, cₙ₋ₖ = −tr(A·Mₖ)/k, Mₖ₊₁ = A·Mₖ + cₙ₋ₖI; det A = (−1)ⁿc₀). The polynomial shows each coefficient rounded to 10 significant figures, and leaves out only terms whose exact coefficient is 0 (or too small to write as a double, under about 10⁻³⁰⁸).

Numbers in text (vectors, complex values, the polynomial) are rounded to 10 significant figures. A value within 10⁻¹² of 0 (relative to the largest entry of the matrix for eigenvalues) is shown as 0. Numbers of 10²¹ or more, or under 10⁻⁶ (other than 0), are written in e-notation, for example 1.4e+22 or 1.5e-7.

Order. Eigenvalues are sorted by real part, then by imaginary part; two real parts within 10⁻⁹ × the largest entry count as equal, so a conjugate pair always shows a − bi before a + bi.

Repeated eigenvalues. Floating-point QR splits a k-fold eigenvalue of a non-diagonalizable (defective) matrix into k values about ε^(1/k) times the matrix size apart (ε ≈ 2.2 × 10⁻¹⁶), sometimes with tiny imaginary parts (Golub and Van Loan, section 7.2.2). So the calculator groups eigenvalues within 1% of the largest entry of each other, rounds the group's mean to 10 significant figures, and checks in exact fractions whether that number is a root of the characteristic polynomial as many times as the group has members. If it is, every member shows that exact value: for [4, 1, 3; 2, 6, 3; −4, −6, −4], det(λI − A) = (λ − 2)³, and the page shows 2, 2, 2. Otherwise the QR values are shown as they are, and a repeated eigenvalue of a defective matrix may then be accurate to only about 16 ÷ k significant figures.

Assumptions

  • The matrix is square, 2×2 to 6×6, with real entries.
  • For a repeated eigenvalue of a symmetric matrix, the eigenvectors shown are one orthonormal set for that eigenvalue; any other orthonormal set of the same eigenspace is also correct. For a non-symmetric matrix with a repeated eigenvalue found exactly (above), the eigenvectors come from the exact null space of A − λI (its RREF): one basis vector per repeat while there are independent ones, then the last basis vector again. A defective eigenvalue has fewer independent eigenvectors than repeats, so the same vector shows more than once. The page then adds a note, for example "λ = 1 repeats twice but has only 1 independent eigenvector. So A is not diagonalizable, and the same eigenvector is shown again for the missing ones." Several defective eigenvalues are listed in ascending order of the eigenvalue, separated by "; ". A repeated eigenvalue that is not found exactly shows the vectors inverse iteration gives, which may differ between repeats.
  • Old links from the previous version of this page (size and cells m00 to m55) still open the same matrix. A missing cell counts as 0.

Worked examples by hand

The default matrix A = [1, 2; 0, 3].

  • Trace 1 + 3 = 4, determinant 1 × 3 − 2 × 0 = 3, so p(λ) = λ² − 4λ + 3 = (λ − 1)(λ − 3). The eigenvalues are 1 and 3.
  • λ = 1: A − I = [0, 2; 0, 2], so 2y = 0 and v = (1, 0).
  • λ = 3: A − 3I = [−2, 2; 0, 0], so x = y and v = (1, 1)/√2 = (0.7071067812, 0.7071067812).

A diagonal matrix diag(1, 2, 3): det(λI − A) = (λ − 1)(λ − 2)(λ − 3), so the eigenvalues are 1, 2, 3, with the unit vectors (1, 0, 0), (0, 1, 0), (0, 0, 1). The previous version of this page showed 0, 0.3333, 0.5, 1 here, which was wrong.

A = [4, 1; 2, 3]. p(λ) = λ² − 7λ + 10 = (λ − 2)(λ − 5).

  • λ = 2: A − 2I = [2, 1; 2, 1], so y = −2x and v = (1, −2)/√5. Its largest component, −2/√5, is made positive: (−0.4472135955, 0.894427191).
  • λ = 5: A − 5I = [−1, 1; 2, −2], so x = y and v = (0.7071067812, 0.7071067812).

A rotation by 90°, A = [0, −1; 1, 0]: p(λ) = λ² + 1, so λ = −i and i. No real vector keeps its direction under a rotation.

A = [2, 1, 0; 1, 2, 1; 0, 1, 2]. Expanding along the first row, det(λI − A) = (λ − 2)((λ − 2)² − 1) − (λ − 2) = (λ − 2)(λ² − 4λ + 2) = λ³ − 6λ² + 10λ − 4. The roots are 2 − √2 = 0.5857864376, 2, and 2 + √2 = 3.414213562. For λ = 2, (A − 2I)v = 0 gives y = 0 and x = −z, so v = (1, 0, −1)/√2 = (0.7071067812, 0, −0.7071067812).

A shear A = [1, 1; 0, 1]: p(λ) = (λ − 1)², so λ = 1, 1. A − I = [0, 1; 0, 0], so (A − I)v = (v₂, 0) = 0 and every eigenvector is a multiple of (1, 0). The page shows [1, 0] for both repeats.

diag(2, 3, 5, 7): p(λ) = (λ − 2)(λ − 3)(λ − 5)(λ − 7) = λ⁴ − 17λ³ + 101λ² − 247λ + 210: 17 = 2 + 3 + 5 + 7, 101 is the sum of the products of pairs (6 + 10 + 14 + 15 + 21 + 35), 247 the sum of the products of triples (30 + 42 + 70 + 105), and 210 = 2 × 3 × 5 × 7, the determinant. The eigenvalues are 2, 3, 5, 7.

Other questions people ask

What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are fundamental concepts in linear algebra. An eigenvalue (λ) is a scalar that, when multiplied by a vector, produces the same result as multiplying the vector by a matrix. An eigenvector is a non-zero vector that, when multiplied by a matrix, results in a scalar multiple of itself. Mathematically, for a matrix A and eigenvector v: Av = λv.

How do I find eigenvalues and eigenvectors?

To find eigenvalues and eigenvectors: 1) Calculate the characteristic polynomial det(A - λI) = 0, 2) Solve for the roots of this polynomial to get eigenvalues, 3) For each eigenvalue, solve the system (A - λI)v = 0 to find the corresponding eigenvectors. Our calculator automates this process for matrices up to 6x6.

What is the characteristic polynomial?

The characteristic polynomial is det(A - λI), where A is the matrix, λ is the eigenvalue variable, and I is the identity matrix. The roots of this polynomial are the eigenvalues of the matrix. For example, for a 2x2 matrix, the characteristic polynomial is λ² - (trace)λ + (determinant) = 0.

Why are eigenvalues and eigenvectors important?

Eigenvalues and eigenvectors are crucial in many applications: diagonalization of matrices, solving systems of differential equations, principal component analysis (PCA) in data science, quantum mechanics, vibration analysis in engineering, and many other fields in physics, engineering, and computer science.

What is the difference between real and complex eigenvalues?

Real eigenvalues are scalar values that are real numbers, while complex eigenvalues have both real and imaginary parts. Real eigenvalues correspond to real eigenvectors, while complex eigenvalues come in conjugate pairs and correspond to complex eigenvectors. Our calculator shows both, writing a complex eigenvalue as a + bi.

How do I interpret the results from this calculator?

The calculator shows: 1) The eigenvalues (λ values), a repeated eigenvalue once per repeat, 2) A unit eigenvector for each eigenvalue, 3) The characteristic polynomial, with the trace and determinant, and 4) How the result is worked out. Each eigenvalue-eigenvector pair satisfies the equation Av = λv.

What is the multiplicity of an eigenvalue?

The multiplicity of an eigenvalue is the number of times it appears as a root of the characteristic polynomial. Algebraic multiplicity counts repeated roots, while geometric multiplicity is the dimension of the eigenspace (number of linearly independent eigenvectors) for that eigenvalue.

Can a matrix have no real eigenvalues?

Yes, some matrices have only complex eigenvalues. For example, rotation matrices in 2D have complex eigenvalues. Our calculator shows complex eigenvalues and their eigenvectors too, for example λ = ±i for the rotation [0, -1; 1, 0].

What is the relationship between eigenvalues and matrix properties?

Eigenvalues reveal important matrix properties: the sum of eigenvalues equals the trace, the product equals the determinant, eigenvalues of A⁻¹ are reciprocals of A's eigenvalues, and eigenvalues of Aⁿ are the nth powers of A's eigenvalues. Zero eigenvalues indicate singular matrices.

How are eigenvalues used in diagonalization?

A matrix is diagonalizable if it has n linearly independent eigenvectors (where n is the matrix size). The diagonal matrix contains eigenvalues on the diagonal, and the transformation matrix contains eigenvectors as columns. This simplifies matrix operations and powers.

What are some practical applications of eigenvalues?

Eigenvalues are used in: structural analysis (natural frequencies), image processing (singular value decomposition), machine learning (principal component analysis), quantum mechanics (energy levels), population dynamics, and many other scientific and engineering applications.

How accurate are the calculations?

Our calculator uses numerical methods (the Jacobi method for symmetric matrices and the QR algorithm for others) and shows results to 10 significant figures. Repeated eigenvalues that are exact roots of the characteristic polynomial are shown exactly; other repeated eigenvalues of non-diagonalizable matrices are less accurate (see How it works). For exact symbolic calculations, specialized computer algebra systems may be preferred. The accuracy is sufficient for most practical applications in science and engineering.