Eigenvalue and Eigenvector Calculator

How to use this eigenvalue and eigenvector calculator

What this calculator does. This tool solves the eigenvalue problem A·v = λ·v for a real square matrix A of order 2 or 3. It reports every eigenvalue, including complex values, and gives a normalized basis for each corresponding right eigenspace. It also reports the matrix Trace and Determinant, two useful invariants tied to the eigenvalues. The calculator is intended for checking linear-algebra work, exploring small matrices, and preparing reproducible examples; it does not replace a full numerical linear-algebra package for large, ill-conditioned, or highly sensitive problems. MIT OpenCourseWare provides a concise explanation of the underlying eigenvalue equation and its geometric meaning.

When to use it. Use the calculator to verify homework computations for 2×2 or 3×3 matrices, inspect whether a real matrix produces complex eigenvalues, compare eigendirections after changing one matrix entry, or check trace/determinant identities before using an eigendecomposition in a larger calculation.

How to calculate. The calculator opens with a complete demonstration matrix, so the first results and the example XLSX workbook are immediately available.

1. Choose Matrix size. Keep 2 × 2 for four entries, or select 3 × 3 to expose all nine matrix positions.
2. Replace the demonstration numbers in the matrix fields. Results update live after every valid edit. Read Eigenvalues first, then use Trace, Determinant, and the Eigenvector basis details table to inspect the full solution.
3. Select Download Excel to export the current validated matrix and results. Reset clears the demonstration and all matrix data; after Reset, Excel export stays disabled until the active matrix is complete and valid again.

Input guide. Matrix size is a required selector with values 2 × 2 or 3 × 3; changing it changes how many entries are required. The fields A11, A12, A21, and A22 are the required 2×2 entries. For a 3×3 matrix, A13, A23, A31, A32, and A33 become required as well. Every matrix entry is a real decimal number with no physical unit. Accepted examples include 2, -0.5, and .25. Use a period for decimals. Do not paste grouping commas, decimal commas, percent signs, currency symbols, or scientific notation such as 1e3. There is no artificial magnitude cap on an individual finite entry, but an edit is rejected if the derived calculation overflows or becomes non-finite. A common mistake is transposing entries: A12 is row 1, column 2, while A21 is row 2, column 1. Increasing one entry does not have a universal one-direction effect on the spectrum; eigenvalues depend on the whole matrix, so interpret changes structurally rather than as a simple “higher input means higher output” rule.

Output guide. Eigenvalues lists the roots λ of det(λI – A) = 0. Real and imaginary parts are shown directly; a zero imaginary part means the eigenvalue is real. Trace is the sum of diagonal entries and, counting multiplicity, equals the sum of the eigenvalues. Determinant is the signed volume-scaling factor and equals the product of the eigenvalues. In the Eigenvector basis details table, Eigenvalue λ identifies a distinct eigenvalue, Algebraic multiplicity states how many times it occurs as a root of the characteristic polynomial, and Eigenspace basis lists one or more independent normalized vectors satisfying A·v = λ·v. A zero eigenvalue means the matrix is singular. Repeated eigenvalues can have one or several independent eigenvectors; therefore multiplicity alone does not determine the eigenspace dimension.

Worked example. The startup matrix is A = [2, 1], [1, 2]