# What are the eigenvectors?

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.

- Page: https://www.acalculator.org/math/eigenvector-calculator
- JSON spec: https://www.acalculator.org/math/eigenvector-calculator.json
- Version: 394f45914fa6

## Default answer

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

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| m | Matrix A | A square matrix, 2×2 to 6×6. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| values | Eigenvalues | Every eigenvalue, repeated ones included, in ascending order of real part and then imaginary part. |
| lambda1 | λ1 | Eigenvalue 1: a number, or text such as 1 + 2i. |
| vector1 | v1 | A unit eigenvector for λ1. |
| lambda2 | λ2 | Eigenvalue 2: a number, or text such as 1 + 2i. |
| vector2 | v2 | A unit eigenvector for λ2. |
| lambda3 | λ3 | Eigenvalue 3: a number, or text such as 1 + 2i. |
| vector3 | v3 | A unit eigenvector for λ3. |
| lambda4 | λ4 | Eigenvalue 4: a number, or text such as 1 + 2i. |
| vector4 | v4 | A unit eigenvector for λ4. |
| lambda5 | λ5 | Eigenvalue 5: a number, or text such as 1 + 2i. |
| vector5 | v5 | A unit eigenvector for λ5. |
| lambda6 | λ6 | Eigenvalue 6: a number, or text such as 1 + 2i. |
| vector6 | v6 | A unit eigenvector for λ6. |
| note | Note | Shown for a matrix that is not diagonalizable: a repeated eigenvalue with fewer independent eigenvectors than repeats. |
| polynomial | Characteristic polynomial | det(λI − A), whose roots are the eigenvalues. |
| trace | Trace | The sum of the diagonal, equal to the sum of the eigenvalues. |
| determinant | Determinant | det(A), equal to the product of the eigenvalues. |
| steps | Working | How the eigenvalues and eigenvectors are found. |

## 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.

## Assumptions

- 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.

## Worked examples

1. m = 1 or 2 or 0 or 3 gives values = 1, 3, lambda1 = 1, lambda2 = 3, vector1 = [1, 0], vector2 = [0.7071067812, 0.7071067812], 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. m = 1 or 0 or 0 or 2 gives values = 1, 2, 3, vector1 = [1, 0, 0], vector2 = [0, 1, 0], vector3 = [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. m = 4 or 1 or 2 or 3 gives values = 2, 5, vector1 = [-0.4472135955, 0.894427191], vector2 = [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. m = 0 or -1 or 1 or 0 gives values = -i, i, lambda1 = -i, lambda2 = i, 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. m = 2 or 1 or 1 or 2 gives lambda1 = 0.585786, lambda2 = 2, lambda3 = 3.414214, vector2 = [0.7071067812, 0, -0.7071067812], 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. m = 1 or 1 or 0 or 1 gives values = 1, 1, vector1 = [1, 0], vector2 = [1, 0], 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. m = 2 or 0 or 0 or 3 gives values = 2, 3, 5, 7, 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.

## FAQ

### 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.

## Sources

- Gilbert Strang, Introduction to Linear Algebra, 5th ed., §6.1 Introduction to Eigenvalues. https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_6-1.pdf