Eigenvector Calculator

Q: Why can't the zero vector be an eigenvector?

The zero vector fits Av = λv for every number λ, because both sides come out as zero. That would make every number an eigenvalue, which tells you nothing useful. So the rule says an eigenvector must be a non-zero vector. Note the difference: an eigen value is allowed to be 0, but an eigen vector is never all zeros.

Q: What is the fast way to find eigenvalues of a 2x2 matrix?

Use this shortcut: λ² − (trace)λ + det = 0 . For the matrix [[a, b], [c, d]], the trace is a + d and the determinant is ad − bc. Example: [[4, 1], [2, 3]] has trace 7 and determinant 10. So λ² − 7λ + 10 = 0, which factors to (λ − 5)(λ − 2) = 0. The eigenvalues are 5 and 2.

Q: Do non-square matrices have eigenvalues?

No. The rule Av = λv needs Av to be the same size as v, and only a square matrix does that. A 2×3 matrix turns a 3-entry vector into a 2-entry vector, so they can never match. For rectangular matrices, people use singular values instead, found with the singular value decomposition (SVD).

Eigenvector Calculator

Introduction

This eigenvector calculator finds the eigenvalues and eigenvectors of any square matrix, from 1×1 up to 8×8. Type your numbers into the grid, press Calculate, and you get the answer with every step shown.

An eigenvector is a special vector. When you multiply it by a matrix, it does not change direction. It only gets longer or shorter. The number that tells you how much it stretches is called the eigenvalue. In math symbols, we write this as Av = λv.

The calculator shows you the characteristic polynomial, the roots of that polynomial (your eigenvalues), and the row reduction work that gives each eigenvector. You can pick the characteristic polynomial method or the direct row reduction method. Each step has an "Explain this step" button if you want to know why it works.

You can enter plain numbers, fractions like 1/3, square roots like sqrt(2), or even complex numbers with i. Results can show as decimals or exact fractions. For 2×2 and 3×3 matrices with real answers, a graph draws the eigenvector directions so you can see what is happening.

There is also a check tool. Enter your own vector and the calculator will tell you if it is a true eigenvector. That makes it handy for homework, class study, and linear algebra practice.

How to use our Eigenvector Calculator

Type the numbers of a square matrix into the grid, then press Calculate. The tool gives you the characteristic polynomial, the eigenvalues, the eigenvectors, step-by-step work, and a plot.

What do you want to compute: Pick "Eigenvalues & Eigenvectors" for the full answer, "Characteristic Polynomial only" for just the polynomial, or "Eigenvalues only" for just the roots.

Rows: Enter how many rows your matrix has, from 1 to 8.

Columns: Enter how many columns your matrix has, from 1 to 8. It must match the rows, since only square matrices have eigenvalues.

Size picker: Click a square in the small grid to set the matrix size fast, up to 8×8.

Matrix A cells: Type one value in each box. You can use whole numbers, decimals, fractions like 1/3, roots like sqrt(2), pi, e, or i. Type I to fill the identity matrix. You can also paste rows from a spreadsheet.

Plus and minus buttons: Use them to add or remove a row or a column without retyping your matrix.

Math keyboard: Open it to tap symbols like √, ^, π, and i straight into the box you clicked last.

Image upload: Add a photo or picture of your matrix to read the numbers from, then type them into the grid.

Examples: Click any example to load a ready-made matrix and see a full solution.

Show results as exact fractions: Turn this on for fractions like 1/3. Leave it off for decimals.

Precision mode: Choose significant figures or decimal places for your answers.

Digits: Pick how many digits to show, from 1 to 15.

Complex number format: Choose rectangular (a + bi) or polar (r·e^(iθ)) for complex eigenvalues.

Solution method: In the steps card, pick "Characteristic Polynomial" or "Direct Substitution / Row Reduction" to change how the work is shown.

Verify Your Answer: Open it, type your own eigenvector, choose an eigenvalue, and press Check to see if Av = λv.

Calculate and Reset: Press Calculate to solve. Press Reset to start over with the default matrix.

What Are Eigenvalues and Eigenvectors?

A matrix can move a vector. It can turn the vector, stretch it, or shrink it. But some special vectors never turn. They only get longer or shorter, or flip to point the other way. Those special vectors are called eigenvectors. The number that tells how much the vector was stretched is called an eigenvalue.

The Main Rule

Eigenvectors and eigenvalues follow one simple equation:

Av = λv

A is a square matrix (same number of rows and columns).
v is the eigenvector. It can never be all zeros.
λ (lambda) is the eigenvalue, just a single number.

In plain words: multiplying the matrix by the vector gives the same answer as multiplying the vector by one number.

How You Find Them

Subtract λ down the diagonal. Build the matrix A − λI, where I is the identity matrix.
Take the determinant and set it to zero. This gives det(A − λI) = 0, called the characteristic equation. The left side is the characteristic polynomial.
Solve for λ. The roots of that polynomial are the eigenvalues. An n×n matrix has n of them, counting repeats.
Plug each λ back in. Row-reduce (A − λI)v = 0 and solve. The answers are the eigenvectors for that eigenvalue.

Multiplicity: Two Kinds

Algebraic multiplicity is how many times an eigenvalue shows up as a root of the polynomial. Geometric multiplicity is how many independent eigenvectors that eigenvalue actually gives you.

When the geometric number is smaller than the algebraic number, the matrix is called defective. A defective matrix cannot be diagonalized, because it does not have enough independent eigenvectors to fill the space.

Complex Eigenvalues

Some real matrices have eigenvalues with an imaginary part, like 2 + 3i. This happens when the matrix rotates vectors, so no real vector keeps its direction. For a real matrix, these always come in pairs: if a + bi is an eigenvalue, then a − bi is one too. Their eigenvectors are pairs as well.

Quick Facts That Help You Check Your Work

Add up all the eigenvalues, and you get the trace (the sum down the diagonal of A).
Multiply all the eigenvalues, and you get the determinant of A.
If one eigenvalue is 0, the matrix has no inverse.
A triangular or diagonal matrix has its eigenvalues sitting right on the diagonal.
A symmetric matrix with real entries always has real eigenvalues and a full set of eigenvectors.
Any eigenvector can be scaled. If v works, then 2v, −v, and 0.5v all work with the same eigenvalue.

Why They Matter

Eigenvalues and eigenvectors show the hidden directions inside a transformation. Engineers use them to find how bridges and buildings shake. Google's PageRank used them to rank web pages. Data scientists use them in Principal Component Analysis (PCA) to shrink big data sets. Physicists use them in quantum mechanics to find energy levels. Computer graphics uses them for scaling and shape analysis.

Formulas used

Eigenvalue–eigenvector equation

Av = \lambda v,\qquad v \neq 0

Characteristic equation (homogeneous form)

(A - \lambda I)v = 0 \iff \det(A - \lambda I) = 0

Characteristic polynomial

p(\lambda) = \det(A - \lambda I) = \sum_{k=0}^{n} c_k \lambda^k

2x2 determinant expansion

\det(A - \lambda I) = (a_{11} - \lambda)(a_{22} - \lambda) - a_{12}a_{21}

3x3 cofactor expansion along the first row

\det(A - \lambda I) = \sum_{j=1}^{n} (-1)^{1+j} a_{1j} M_{1j}

Eigenvalues of a 2x2 matrix (quadratic formula)

\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\qquad p(\lambda) = a\lambda^2 + b\lambda + c

Factored characteristic polynomial and multiplicities

\det(A - \lambda I) = \prod_{k=1}^{m} (\lambda_k - \lambda)^{a_k},\qquad g_k = n - \operatorname{rank}(A - \lambda_k I) \le a_k

Complex eigenvalue in polar form

\lambda = a + bi = r\,e^{i\theta},\qquad r = \sqrt{a^2 + b^2},\ \ \theta = \operatorname{atan2}(b, a)

Frequently asked questions

Why can't the zero vector be an eigenvector?

The zero vector fits Av = λv for every number λ, because both sides come out as zero. That would make every number an eigenvalue, which tells you nothing useful. So the rule says an eigenvector must be a non-zero vector.

Note the difference: an eigenvalue is allowed to be 0, but an eigenvector is never all zeros.

How many eigenvectors does a matrix have?

Each eigenvalue has endless eigenvectors, but only a few real directions. If v works, then 2v, -v, and 0.5v all work with the same eigenvalue.

What counts is the number of independent directions. An n×n matrix has at most n independent eigenvectors. If it has fewer, the matrix is called defective.

What is an eigenspace?

An eigenspace is the set of all eigenvectors for one eigenvalue, plus the zero vector. In math terms, it is the null space of A − λI.

One independent eigenvector = the eigenspace is a line.
Two independent eigenvectors = it is a plane.

The dimension of the eigenspace is the geometric multiplicity of that eigenvalue.

How do you normalize an eigenvector?

Divide every entry by the length of the vector. The length is the square root of the sum of the squares of the entries.

Example: the vector [3, 4] has length √(9 + 16) = 5. Divide each part by 5 to get [0.6, 0.8].

Normalizing does not change the eigenvalue or the direction. Textbooks and PCA often ask for these unit eigenvectors.

What does a negative eigenvalue mean?

It means the matrix flips the eigenvector so it points the opposite way, and may stretch or shrink it too.

λ = −1: the vector flips only.
λ = −2: the vector flips and doubles in length.
λ = −0.5: the vector flips and gets half as long.

In physics and engineering models, negative eigenvalues often point to decay or instability.

What does it mean when an eigenvalue is zero?

It means the matrix squashes that eigenvector down to nothing: Av = 0.

When 0 is an eigenvalue, the determinant of the matrix is also 0. So the matrix is singular, it has no inverse, and its columns are not independent. The eigenvectors for λ = 0 are exactly the null space of A.

What is the fast way to find eigenvalues of a 2x2 matrix?

Use this shortcut: λ² − (trace)λ + det = 0. For the matrix [[a, b], [c, d]], the trace is a + d and the determinant is ad − bc.

Example: [[4, 1], [2, 3]] has trace 7 and determinant 10. So λ² − 7λ + 10 = 0, which factors to (λ − 5)(λ − 2) = 0. The eigenvalues are 5 and 2.

Are eigenvectors of a symmetric matrix always perpendicular?

Yes, when the eigenvalues are different. For a real symmetric matrix, eigenvectors from different eigenvalues always meet at 90 degrees, so their dot product is 0.

If an eigenvalue repeats, you can still pick perpendicular eigenvectors inside its eigenspace. This is why symmetric matrices are used so much in PCA and statistics.

How do you know if a matrix is diagonalizable?

A matrix is diagonalizable if it has a full set of n independent eigenvectors. That happens when every eigenvalue's geometric multiplicity equals its algebraic multiplicity.

Two quick checks:

If all n eigenvalues are different, it is diagonalizable.
Every real symmetric matrix is diagonalizable.

When it works, you can write A = PDP⁻¹, where D holds the eigenvalues and P holds the eigenvectors.

Do non-square matrices have eigenvalues?

No. The rule Av = λv needs Av to be the same size as v, and only a square matrix does that. A 2×3 matrix turns a 3-entry vector into a 2-entry vector, so they can never match.

For rectangular matrices, people use singular values instead, found with the singular value decomposition (SVD).

What are the eigenvalues of an inverse matrix?

If λ is an eigenvalue of A, then 1/λ is an eigenvalue of A⁻¹, and the eigenvector stays the same. This only works when no eigenvalue is 0, since a matrix with a zero eigenvalue has no inverse.

Related rules: A² has eigenvalues λ², A^k has λ^k, and A + cI has eigenvalues λ + c. The eigenvectors do not change in any of these.

Does a matrix have the same eigenvalues as its transpose?

Yes. A and Aᵀ have the exact same characteristic polynomial, so they share the same eigenvalues with the same multiplicities.

Their eigenvectors are usually different though. The eigenvectors of Aᵀ are called the left eigenvectors of A.

How can you check if a vector is an eigenvector?

Multiply the matrix by the vector. If the answer is just the same vector scaled by one number, it is an eigenvector, and that number is the eigenvalue.

A quick test: divide each entry of Av by the matching entry of v. If every ratio is the same number, you have an eigenvector. If the ratios differ, you do not.

Can a 3x3 matrix with real numbers have all complex eigenvalues?

No. Complex eigenvalues of a real matrix always come in pairs, like 2 + 3i together with 2 − 3i. A 3×3 matrix has three eigenvalues, so after one pair there is always one left over that must be real.

In general, any real matrix of odd size has at least one real eigenvalue.

What does the word eigen mean?

Eigen is a German word meaning "own" or "characteristic". So an eigenvector is the matrix's own special vector, and an eigenvalue is its own special number.

It is said like "EYE-gun". That is also why det(A − λI) = 0 is called the characteristic equation.

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