Eigenvalue Eigenvector Calculator - 2x2 and 3x3 Eigenvalue Calculator
Eigenvalue eigenvector calculator for 2x2 and 3x3 matrices that returns real eigenvalues, matching eigenvectors, the trace, and the characteristic polynomial.
Eigenvalue Eigenvector Calculator
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What Is Eigenvalue Eigenvector Calculator?
An eigenvalue eigenvector calculator turns a 2x2 or 3x3 matrix into its real eigenvalues and matching eigenvectors in one pass. The page solves det(A - lambda * I) = 0 for real roots and pairs each with a normalized eigenvector satisfying A * v = lambda * v.
- • Linear algebra homework: Verify a hand-computed characteristic polynomial and the trace-determinant quadratic for a 2x2 matrix.
- • Stability and vibration: Read the largest real eigenvalue off a small system to judge long-run growth or label a vibration mode.
- • Principal component analysis: Cross-check the first eigenvector of a 2x2 or 3x3 covariance matrix before downstream PCA work.
- • Markov chain steady state: Confirm that lambda = 1 is an eigenvalue of a transition matrix and read off the matching eigenvector.
For a 2x2 matrix the trace-determinant quadratic lambda^2 - tr(A) lambda + det(A) = 0 gives the eigenvalues from the trace and determinant. For a 3x3 matrix the calculator reads off the cubic characteristic polynomial and applies a real-root solver, so the result panel shows the three real roots when they exist and a single real root with a complex-conjugate note otherwise.
The calculator only returns real eigenvectors. When a matrix has a complex eigenvalue, the page prints the conjugate pair.
When the same 2x2 or 3x3 matrix also needs its cofactor matrix or its inverse through the adjugate identity, adjoint matrix calculator sits one step away in the same math-conversion folder.
How Eigenvalue Eigenvector Calculator Works
The calculator is a driver for the eigenequation A v = lambda v. It reads the matrix, builds the characteristic polynomial det(A - lambda I), solves it for real roots, and pairs each root with a real eigenvector from the null space of (A - lambda I).
- A: The 2x2 or 3x3 input matrix with real entries.
- lambda: An eigenvalue, the scalar that scales the eigenvector when A acts on it.
- v: An eigenvector in the null space of A - lambda I; returned normalized to length 1.
- tr(A): Sum of the diagonal entries of A, equal to the sum of the eigenvalues for a 2x2 matrix.
- det(A): Determinant of A, equal to the product of the eigenvalues for a 2x2 matrix.
For a 2x2 matrix the trace-determinant quadratic is the cleanest path, with l1 = tr(A)/2 + sqrt(tr(A)^2/4 - det(A)) and l2 = tr(A)/2 - sqrt(tr(A)^2/4 - det(A)). For a 3x3 matrix the page uses a real-root cubic solver and labels the complex pair in the complex-eigenvalue note when only one real root exists.
Worked 2x2 example: A = [[2, 1], [1, 2]]
a = 2, b = 1, c = 1, d = 2. A symmetric 2x2 matrix.
tr(A) = 4, det(A) = 3. The polynomial is lambda^2 - 4*lambda + 3 = (lambda - 1)(lambda - 3) = 0.
lambda_1 = 3, lambda_2 = 1, v_1 = (0.7071, 0.7071), v_2 = (0.7071, -0.7071).
A stretches the (1, 1) direction by 3 and the (1, -1) direction by 1. The eigenvectors are orthogonal, the expected behavior for a real symmetric matrix.
Worked 3x3 example: A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]
a11 = 1, a12 = 2, a13 = 3, a21 = 0, a22 = 1, a23 = 4, a31 = 5, a32 = 6, a33 = 0.
tr(A) = 2, det(A) = 1, characteristic polynomial lambda^3 - 2*lambda^2 - 23*lambda - 1 = 0. Three real roots, dominant near 5.812.
lambda_1 = 5.8121, lambda_2 = -3.3434, lambda_3 = -0.4686, with one normalized real eigenvector each.
The dominant eigenvalue is positive and largest in magnitude, so repeated multiplication by A pushes any starting vector toward the v_1 direction.
According to Wikipedia, an eigenvector of an n x n matrix A is a nonzero vector v such that A v = lambda v, and lambda is a root of the characteristic polynomial det(A - lambda I) = 0.
According to MIT OpenCourseWare 18.06 (Strang), for a 2x2 matrix the eigenvalues are lambda_1 = tr(A)/2 + sqrt(tr(A)^2/4 - det(A)) and lambda_2 = tr(A)/2 - sqrt(tr(A)^2/4 - det(A)).
For a real symmetric positive-definite matrix the same 2x2 or 3x3 system factors as A = L L^T through a triangular decomposition, which is exactly what the Cholesky decomposition calculator returns.
Key Concepts Explained
Four ideas from linear algebra cover the entire pipeline the calculator runs, and they are the same terms that show up in any first course on eigenvalues and eigenvectors.
Eigenequation A v = lambda v
The defining equation of an eigenpair. A acts on a nonzero vector v and the result is parallel to v, scaled by lambda.
Characteristic polynomial det(A - lambda I)
A scalar polynomial in lambda whose roots are the eigenvalues. A 2x2 input gives a quadratic and a 3x3 input gives a cubic.
Trace and determinant for a 2x2 matrix
For a 2x2 matrix, the trace is lambda_1 + lambda_2 and the determinant is lambda_1 * lambda_2. Those two numbers are all you need to solve the quadratic.
Real symmetric matrices have real eigenvalues
A real symmetric matrix A = A^T has only real eigenvalues, and the eigenvectors for distinct eigenvalues are orthogonal.
These four concepts are the entire vocabulary the calculator uses.
Because the eigenvalue calculator normalizes each real eigenvector to length 1, the matching 2D or 3D length is always 1, and the vector magnitude calculator is the quickest way to confirm that.
How to Use This Calculator
Pick the matrix size, type the entries, and read the real eigenvalues and eigenvectors in the right-hand panel. The eigenvalue eigenvector calculator updates the result panel on every keystroke.
- 1 Choose the matrix size: Select 2x2 or 3x3. The matching entry grid appears below.
- 2 Type the matrix entries: Real numbers, decimals allowed. The result panel updates on every keystroke.
- 3 Read the trace, determinant, and characteristic polynomial: The first three rows of the result panel.
- 4 Read the real eigenvalues: lambda_1 is the dominant real root, lambda_2 the second, lambda_3 the third for a 3x3 matrix.
- 5 Read the matching eigenvectors: Each real eigenvalue is paired with a normalized eigenvector. A blank cell means a complex eigenvalue.
- 6 Reset to the example matrix: Reset restores the default 2x2 symmetric example (eigenvalues 3 and 1).
Suppose a 2x2 covariance matrix is A = [[4, 1], [1, 3]]. Type the four entries, then read lambda_1 = 4.618, lambda_2 = 2.382, with orthogonal eigenvectors. The dominant eigenvalue is the variance along the principal component.
For a real symmetric matrix the eigenvectors for distinct eigenvalues are orthogonal, and the dot product calculator is the cleanest way to confirm that the dot product of v_1 and v_2 is zero.
Benefits of Using This Calculator
The eigenvalue eigenvector calculator keeps the entire eigenpipeline on screen at once, so the user sees the polynomial, eigenvalues, and eigenvectors together.
- • Real eigenvalues and eigenvectors in one tool: A size selector swaps between 2x2 and 3x3, the two matrix sizes that show up in homework and small PCA workflows.
- • Characteristic polynomial shown as a string: The result panel includes the characteristic polynomial in lambda, so the user can check it against a hand expansion.
- • Trace and determinant paired with the eigenvalues: For a 2x2 matrix the trace equals lambda_1 + lambda_2 and the determinant equals lambda_1 * lambda_2.
- • Clear complex-eigenvalue message: When the matrix has no real eigenvalues (a 2x2 rotation) the page shows the conjugate pair and explains that no real eigenvector exists.
- • Decimal-friendly arithmetic with normalization: Entries can be decimals, and each eigenvector is returned as a normalized vector for downstream use.
The right-hand panel mirrors the textbook sequence: trace, determinant, characteristic polynomial, eigenvalues, eigenvectors. That visual structure is the fastest way to learn how the trace-determinant quadratic fits into the 2x2 case.
The cross product of the two matching 3D eigenvectors is perpendicular to both, and the cross product calculator returns that vector and its magnitude in one step.
Factors That Affect Your Results
A handful of matrix properties decide whether the eigenvalues come out real, repeated, or complex.
Symmetry of the matrix
A real symmetric matrix A = A^T always has real eigenvalues, and the eigenvectors for distinct eigenvalues are orthogonal.
Sign of the discriminant (2x2)
tr(A)^2 - 4 det(A) above zero means two distinct real eigenvalues, exactly zero means a repeated real eigenvalue, below zero means a complex conjugate pair with no real eigenvector.
Diagonal and triangular matrices
A diagonal matrix has the diagonal entries as its eigenvalues and the standard basis vectors as its eigenvectors. A triangular matrix shares those diagonal eigenvalues, but its eigenvectors are shifted by the off-diagonal terms.
Zero as an eigenvalue
When the matrix is singular, det(A) = 0, and lambda = 0 is one of the eigenvalues. The matching eigenvector is in the null space of A.
Real roots of a 3x3 cubic
A 3x3 characteristic polynomial is a cubic, so it can have one real root (with a complex conjugate pair) or three real roots.
- • The calculator is restricted to 2x2 and 3x3 square matrices. For 4x4 and larger the characteristic polynomial is a quartic or higher, and a numerical library is the practical option.
- • The display is rounded to four decimal places, so a cubic root that is algebraically irrational shows up as a decimal. Keep full double precision through the calculation and round only the final displayed values.
- • Only real eigenvectors are returned. A real matrix with complex eigenvalues has no real eigenvector, so the result panel shows the conjugate pair and leaves the eigenvector cells blank.
These factors and limitations explain when the result panel is exact and when it is rounded or partial.
According to MIT OpenCourseWare 18.06 (Strang), for a 2x2 matrix the eigenvalues satisfy lambda^2 - tr(A) lambda + det(A) = 0, and the eigenvectors of a real symmetric matrix for distinct eigenvalues are orthogonal.
When the same 2x2 or 3x3 matrix also needs the full characteristic polynomial written out as an explicit expression in lambda, the characteristic polynomial calculator sits one step away in the same math-conversion folder.
Frequently Asked Questions
Q: How do I find eigenvalues and eigenvectors of a 2x2 matrix?
A: For a 2x2 matrix A, write the trace tr(A) and the determinant det(A), then solve the trace-determinant quadratic lambda^2 - tr(A) lambda + det(A) = 0. The two real roots are the eigenvalues lambda_1 and lambda_2; for each real root solve (A - lambda I) v = 0 for a nonzero v and normalize to length 1.
Q: How do I find eigenvalues of a 3x3 matrix?
A: For a 3x3 matrix A, form the characteristic polynomial det(A - lambda I) = 0, which is a cubic in lambda. Solve the cubic for its real roots; a 3x3 cubic has either one real root (with a complex conjugate pair) or three real roots. The matching eigenvectors come from the null space of (A - lambda I) for each real root.
Q: How do I find eigenvectors from eigenvalues?
A: Once you have a real eigenvalue lambda, plug it into (A - lambda I) v = 0 and solve the linear system for a nonzero v. The nonzero solutions form the eigenspace for lambda, and any nonzero vector in that space is an eigenvector. The calculator returns a normalized vector of length 1.
Q: How many eigenvalues does a matrix have?
A: An n x n matrix has at most n eigenvalues, counted with multiplicity. Over the real numbers a 2x2 matrix can have 0, 1, or 2 real eigenvalues and a 3x3 matrix can have 1 or 3 real eigenvalues; the missing roots are always complex conjugates.
Q: Are eigenvectors always orthogonal?
A: Not in general. The eigenvectors of a real symmetric matrix with distinct eigenvalues are always orthogonal, but a non-symmetric matrix can have eigenvectors that point in arbitrary directions. Orthogonality is a property of the matrix, not of eigenvectors as a class.
Q: Can zero be an eigenvalue of a matrix?
A: Yes. Zero is an eigenvalue whenever the matrix is singular, meaning det(A) = 0. The matching eigenvector is a nonzero vector in the null space of A, and any vector that A sends to zero is an eigenvector for lambda = 0.