Eigenvalue and Eigenvector Calculator
Calculate eigenvalue eigenvector instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
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Eigenvalues are found by solving the characteristic equation, which sets the determinant of (A - lambda * I) to zero. For a 3x3 matrix, this produces a cubic polynomial. Each eigenvalue lambda has corresponding eigenvectors found by solving the homogeneous system (A - lambda * I)v = 0.
Last reviewed: December 2025
Worked Examples
Example 1: 3x3 Symmetric Matrix Eigenvalues
Example 2: Matrix with Repeated Eigenvalue
Background & Theory
The Eigenvalue and Eigenvector Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Eigenvalue and Eigenvector Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
det(A - lambda * I) = 0 (characteristic equation), then solve (A - lambda * I)v = 0 for eigenvectors
Eigenvalues are found by solving the characteristic equation, which sets the determinant of (A - lambda * I) to zero. For a 3x3 matrix, this produces a cubic polynomial. Each eigenvalue lambda has corresponding eigenvectors found by solving the homogeneous system (A - lambda * I)v = 0.
Worked Examples
Example 1: 3x3 Symmetric Matrix Eigenvalues
Problem: Find eigenvalues and eigenvectors of A = [[2, 1, 0], [1, 3, 1], [0, 1, 2]].
Solution: Characteristic polynomial: lambda^3 - 7*lambda^2 + 14*lambda - 8 = 0\nFactors: (lambda - 1)(lambda - 2)(lambda - 4) = 0\nEigenvalues: lambda = 1, 2, 4\n\nFor lambda = 1: [1, 1, 0; 1, 2, 1; 0, 1, 1]v = 0 -> v1 = [1, -1, 1]\nFor lambda = 2: [0, 1, 0; 1, 1, 1; 0, 1, 0]v = 0 -> v2 = [1, 0, -1]\nFor lambda = 4: [-2, 1, 0; 1, -1, 1; 0, 1, -2]v = 0 -> v3 = [1, 2, 1]
Result: Eigenvalues: 1, 2, 4 | Matrix is symmetric, so orthogonally diagonalizable
Example 2: Matrix with Repeated Eigenvalue
Problem: Find eigenvalues of A = [[3, 0, 0], [0, 3, 0], [0, 0, 5]].
Solution: This is a diagonal matrix, so eigenvalues are the diagonal entries.\nlambda1 = lambda2 = 3 (algebraic multiplicity 2)\nlambda3 = 5\nEigenvectors for lambda = 3: any vector in the xy-plane, v1 = [1,0,0], v2 = [0,1,0]\nEigenvector for lambda = 5: v3 = [0,0,1]\nGeometric multiplicity = algebraic multiplicity, so diagonalizable.
Result: Eigenvalues: 3, 3, 5 | Diagonalizable (geometric multiplicity = algebraic multiplicity)
Frequently Asked Questions
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Each field is labelled with the required unit (metric or imperial). Gather your source values before starting — for example, a weight measurement in kilograms, a distance in metres, or a dollar amount — and enter them exactly as measured. The formula section on this page lists every variable and explains what each represents.
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You may use the results for reference and educational purposes. For professional reports, academic papers, or critical decisions, we recommend verifying outputs against peer-reviewed sources or consulting a qualified expert in the relevant field.
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How accurate are the results from Eigenvalue and Eigenvector Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy