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. 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. The sum of all eigenvalues equals the trace (the sum of diagonal entries), and the product of all eigenvalues equals the determinant.