Eigenvalue Calculator

Q: What size matrices can this calculator handle?

This calculator works with square matrices from 1×1 up to 6×6 . You can also enter non-square matrices, but eigenvalues and eigenvectors only exist for square matrices. The tool will tell you if your matrix is not square.

Q: What does it mean when an eigenvalue is complex?

A complex eigenvalue has an imaginary part, like 3 + 2i . This happens when the characteristic polynomial has no real roots for that factor. Complex eigenvalues always come in conjugate pairs for real matrices, so if 3 + 2i is an eigenvalue, then 3 − 2i is also an eigenvalue.

Q: What is the difference between algebraic and geometric multiplicity?

Algebraic multiplicity is how many times an eigenvalue shows up as a root of the characteristic polynomial. Geometric multiplicity is how many independent eigenvectors belong to that eigenvalue. Geometric multiplicity is always less than or equal to algebraic multiplicity.

Q: How do I switch between decimal and fraction output?

Turn on the Show results as fractions toggle in the Options section. When it is on, the calculator displays results as fractions wherever possible. When it is off, you can set how many decimal places or significant digits to show.

Q: What does positive definite mean?

A matrix is positive definite when all of its eigenvalues are greater than zero. This property is important in optimization and statistics. The calculator checks definiteness for you and shows the result in the Matrix Properties section.

Q: Why does the calculator say my matrix is singular?

A matrix is singular when at least one eigenvalue is zero. A singular matrix has no inverse and its determinant is zero. The calculator detects this by checking if any computed eigenvalue equals zero.

Eigenvalue Calculator

Updated Jul 11, 2026 By Jehan Wadia

Introduction

This eigenvalue calculator finds the eigenvalues and eigenvectors of any matrix up to 6×6. Just type your matrix into the grid, paste it as text, or describe it in plain English — then hit Calculate. The tool solves the characteristic polynomial, shows each eigenvalue with its algebraic and geometric multiplicity, and gives you the matching eigenvectors. Every result comes with a step-by-step solution so you can follow the math from start to finish.

You also get useful matrix properties like the determinant, trace, symmetry, definiteness, and whether the matrix is diagonalizable. A built-in chart plots your eigenvalues on the complex plane so you can see them at a glance. The calculator supports fractions, decimals, constants like pi and e, and even expressions like sqrt(2) or 2^3 inside each cell. Whether you are a student checking homework or someone solving a linear algebra problem, this tool gives you fast, clear answers with full explanations.

How to Use Our Eigenvalue Calculator

Enter a square matrix and this calculator will find its eigenvalues, eigenvectors, characteristic polynomial, and key matrix properties with full step-by-step work shown.

Pick an input mode. Click Grid / Cells to type numbers into a visual matrix grid. Click Raw Text / Paste to type or paste your matrix in bracket format like [[2,1],[1,2]]. Click Natural Language to describe your matrix in plain English.

Set the matrix size. In Grid mode, use the Rows and Columns buttons to change the size of your matrix, or click a preset like 2×2 or 3×3. The maximum size is 6×6.

Enter your matrix values. Type a number, fraction, decimal, or math expression into each cell. You can use values like 1/3, sqrt(2), pi, or 2^5. You can also paste a block of data from a spreadsheet.

Choose your display options. Turn on Show results as fractions if you want fraction output instead of decimals. Use the Precision mode dropdown and Digits field to control how many decimal places or significant digits appear in your results.

Click Calculate. Press the Calculate button to run the computation. Your results will appear below, including the characteristic polynomial, each eigenvalue with its eigenvectors, a diagonalizability check, and a plot of the eigenvalues on the complex plane.

What Are Eigenvalues and Eigenvectors?

An eigenvalue is a special number tied to a matrix. When you multiply a matrix by a certain vector and the result is just that same vector stretched or shrunk by a number, that number is the eigenvalue. The vector that gets stretched or shrunk is called an eigenvector.

Think of it this way: a matrix usually changes a vector's direction and size. But an eigenvector only changes in size, not direction. The eigenvalue tells you how much bigger or smaller it gets. If the eigenvalue is 2, the vector doubles in length. If it is −1, the vector flips and stays the same length.

How to Find Eigenvalues

To find eigenvalues, you solve the characteristic equation. You take the matrix, subtract λ (lambda) times the identity matrix, and set the determinant equal to zero. This gives you a polynomial called the characteristic polynomial. The roots of that polynomial are the eigenvalues.

Once you know each eigenvalue, you plug it back into the equation (A − λI)x = 0 and solve for the vector x. The non-zero solutions are the eigenvectors for that eigenvalue. This involves reducing the matrix to row echelon form to find the null space. For 2×2 matrices, the characteristic polynomial is a quadratic that you can solve with the quadratic formula.

Key Terms

Algebraic multiplicity is how many times an eigenvalue appears as a root of the characteristic polynomial. Geometric multiplicity is how many independent eigenvectors belong to that eigenvalue. If these two numbers match for every eigenvalue, the matrix is diagonalizable, which means it can be broken into simpler parts.

Why Eigenvalues Matter

Eigenvalues show up in many real-world uses. Engineers use them to study vibrations in buildings and bridges. Scientists use them in data analysis to find patterns. They help with image compression, search engine rankings, and quantum physics. Any time a system can be described by a matrix, eigenvalues help reveal how that system behaves.

Eigenvalues also tell you useful facts about a matrix. If any eigenvalue is zero, the matrix is singular and cannot be inverted — you can verify this with our inverse matrix calculator. The sum of all eigenvalues equals the trace (the sum of diagonal entries), and the product of all eigenvalues equals the determinant. For related linear algebra operations, try our matrix multiplication calculator, dot product calculator, or system of equations calculator.

Formulas used

Characteristic Polynomial

p(\lambda) = \det(A - \lambda I) = 0

Characteristic Polynomial (2×2)

p(\lambda) = \lambda^2 - \operatorname{tr}(A)\,\lambda + \det(A)

Faddeev–LeVerrier Recurrence

c_k = -\frac{\operatorname{tr}(A \cdot N_{k-1})}{k}, \quad N_k = A \cdot N_{k-1} + c_k I

Eigenvector (Null Space)

(A - \lambda I)\mathbf{v} = \mathbf{0}

Trace–Eigenvalue Relation

\sum_{i=1}^{n} \lambda_i = \operatorname{tr}(A)

Determinant–Eigenvalue Relation

\prod_{i=1}^{n} \lambda_i = \det(A)

Frequently asked questions

What size matrices can this calculator handle?

This calculator works with square matrices from 1×1 up to 6×6. You can also enter non-square matrices, but eigenvalues and eigenvectors only exist for square matrices. The tool will tell you if your matrix is not square.

Can I enter fractions or square roots in the matrix cells?

Yes. Each cell accepts whole numbers, decimals, fractions like 1/3, expressions like sqrt(2) or 2^3, and constants like pi, e, and i. You can even combine them, for example 2/3 + sqrt(5).

What does it mean when an eigenvalue is complex?

A complex eigenvalue has an imaginary part, like 3 + 2i. This happens when the characteristic polynomial has no real roots for that factor. Complex eigenvalues always come in conjugate pairs for real matrices, so if 3 + 2i is an eigenvalue, then 3 − 2i is also an eigenvalue.

What is the difference between algebraic and geometric multiplicity?

Algebraic multiplicity is how many times an eigenvalue shows up as a root of the characteristic polynomial. Geometric multiplicity is how many independent eigenvectors belong to that eigenvalue. Geometric multiplicity is always less than or equal to algebraic multiplicity.

How do I know if my matrix is diagonalizable?

The calculator checks this for you in Step 5 of the solution. A matrix is diagonalizable when every eigenvalue has its geometric multiplicity equal to its algebraic multiplicity. If any eigenvalue has fewer independent eigenvectors than its algebraic multiplicity, the matrix is not diagonalizable.

Can I paste a matrix from a spreadsheet?

Yes. Copy a block of cells from Excel, Google Sheets, or any spreadsheet, then paste it into any cell in the grid. The calculator will auto-fill all the cells with your data and resize the grid to match.

What does the complex plane chart show?

The chart plots each eigenvalue as a dot. The horizontal axis is the real part and the vertical axis is the imaginary part. If all dots sit on the horizontal axis, every eigenvalue is real. Dots above or below the axis mean the eigenvalue has an imaginary part.

How do I switch between decimal and fraction output?

Turn on the Show results as fractions toggle in the Options section. When it is on, the calculator displays results as fractions wherever possible. When it is off, you can set how many decimal places or significant digits to show.

What does positive definite mean?

A matrix is positive definite when all of its eigenvalues are greater than zero. This property is important in optimization and statistics. The calculator checks definiteness for you and shows the result in the Matrix Properties section.

Why does the calculator say my matrix is singular?

A matrix is singular when at least one eigenvalue is zero. A singular matrix has no inverse and its determinant is zero. The calculator detects this by checking if any computed eigenvalue equals zero.

Can I upload a photo of a matrix?

You can upload a PNG or JPG image using the Upload a Matrix Image section. However, automatic image reading is not available in this tool. Your image will be shown for reference so you can type the values into the grid yourself.

What is the characteristic polynomial?

The characteristic polynomial is the equation you get when you compute det(A − λI) = 0. Its roots are the eigenvalues. The calculator shows you this polynomial in Step 2 of the solution and writes it in both expanded and factored form.

How does the Natural Language input mode work?

Type a request in plain English like "eigenvalues of [[3,1],[0,2]]". The calculator looks for a matrix inside your text, parses it, and shows a preview. Then press Calculate to get the full solution.

What does the trace check in Matrix Properties mean?

The trace of a matrix is the sum of its diagonal entries. A key property in linear algebra is that the sum of all eigenvalues always equals the trace. The calculator verifies this to confirm the results are correct.

Can I use this calculator for a non-square matrix?

You can enter a non-square matrix, but the calculator will tell you that eigenvalues and eigenvectors are only defined for square matrices. You need the same number of rows and columns to compute eigenvalues.

How do I undo a change?

Click the Undo button to go back to your previous matrix. This restores the last grid values, matrix size, and input mode you had before your most recent change.

What method does this calculator use to find eigenvalues?

The calculator uses the Faddeev–LeVerrier algorithm to build the characteristic polynomial, then applies the Durand–Kerner method to find all roots numerically. For eigenvectors, it row-reduces the matrix (A − λI) and computes the null space.

Can I enter repeating decimals?

Yes. Use the format 1.3(56) where the digits in parentheses repeat forever. For example, 0.(3) means 0.333… which equals 1/3. The calculator converts the repeating decimal to an exact value before computing.

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