Eigenvalue & Matrix Properties Calculator: Rank, Condition Number, Stability

The central object on this page is the eigendecomposition. When it exists, a square matrix A factors as A = QΛQ⁻¹, with the eigenvalues sitting on the diagonal of Λ and the eigenvectors as columns of Q. Almost every property of A drops out of that one factorization. The determinant is the product of the eigenvalues. The trace is the sum. The rank, for a normal matrix, is the count of non-zero eigenvalues. The condition number is the ratio of the largest absolute eigenvalue to the smallest. And A^k for any power k is just QΛ^kQ⁻¹, which is why eigendecomposition is the standard language for dynamical systems and the foundation of PCA.

For matrix operations themselves (addition, multiplication, transpose, inverse, RREF, or step-by-step row reduction on a system Ax = b), see the matrix calculator. This page handles the diagnostics: rank, trace, eigenvalues, eigenvectors, singular values, and the condition number κ(A). Two halves of the same workflow, separated because they answer different questions and expect different inputs.

Condition number is the property to actually check before any numerical work. κ near 1 means the matrix is well-behaved and small input errors stay small. κ around 10⁶ means tiny perturbations in b can swing the solution to Ax = b by a factor of a million. Past 10¹² and you're past the resolution of double-precision floats: the digits you compute aren't really digits, they're roundoff noise dressed up as signal.

Sizes are capped at 4×4 on purpose. Eigenvalues for matrices larger than that come out of an iterative algorithm (QR with shifts), and the results are harder to interpret on a single screen. For a 6×6 covariance matrix or a 5×5 Jacobian from an optimization, NumPy's np.linalg.eig and SciPy's scipy.linalg.eig run in milliseconds and produce double-precision answers. This page is for understanding what a matrix is, not for production-scale numerics.

Vectors are single-row or single-column matrices. A column vector (n×1) multiplies naturally with an m×n matrix from the left: Av is m×1. A row vector (1×n) multiplies from the right: vᵀA is 1×m. Choose orientation based on the operation you need.

Enter matrices row by row, separating entries with commas or tabs. For a 3×3 matrix, input three lines with three values each. Most parsers accept scientific notation (2.5e-4) and negative numbers. Avoid trailing commas—some parsers treat them as extra zero entries.

Copy directly from spreadsheets for larger matrices. Tab-delimited pastes from Excel or Google Sheets typically parse correctly. Verify dimensions after pasting: an unexpected 4×5 instead of 4×4 causes cryptic errors downstream.

Rank counts linearly independent rows (equivalently, columns). For an m×n matrix, rank ≤ min(m, n). Full rank means rank equals the smaller dimension—no row is a linear combination of others. Rank deficiency signals redundancy or collapsing transformations.

Trace is the sum of diagonal entries: tr(A) = a₁₁ + a₂₂ + ... + aₙₙ. Only defined for square matrices. The trace equals the sum of eigenvalues—a quick consistency check if you compute both. Trace is invariant under similarity: tr(P⁻¹AP) = tr(A).

Determinant measures signed volume scaling under the transformation A represents. det(A) = 0 means singular (non-invertible); det(A) ≠ 0 means invertible. For 2×2: det = ad − bc. The determinant equals the product of eigenvalues.

An eigenvector v satisfies Av = λv: the matrix stretches v by factor λ without rotating it. Each eigenvalue has at least one associated eigenvector (often infinitely many, forming an eigenspace). Eigenpairs reveal the matrix's fundamental action on space.

Positive eigenvalues mean stretching along that direction; negative eigenvalues flip and stretch; λ = 0 collapses that direction entirely. Complex eigenvalues (for real matrices) come in conjugate pairs and indicate rotational components—the transformation spirals rather than purely stretches.

Applications abound. In differential equations, eigenvalues determine stability (negative real parts = stable). In principal component analysis, the largest eigenvalues correspond to directions of greatest variance. In Markov chains, the eigenvalue 1 identifies the steady-state distribution.

The condition number κ(A) of a matrix is the ratio σ_max(A) / σ_min(A) of its largest to smallest singular values. For a square non-singular A, that's also ‖A‖₂ · ‖A⁻¹‖₂. Mathematically it's a number. Operationally it's a leverage factor. If your input b carries a relative error of ε, the relative error in the solution to Ax = b can be as large as κ(A) · ε. The bound is tight, not pessimistic: there are perturbation directions where the error attains it.

Trefethen and Bau's Numerical Linear Algebra (Lecture 12) frames the rule of thumb cleanly. With double-precision arithmetic (machine epsilon ≈ 10⁻¹⁶) and a condition number κ, you lose roughly log₁₀(κ) digits of accuracy. If κ ≈ 10⁶, you keep ~10 digits of meaningful answer out of 16. If κ ≈ 10¹⁰, you keep about 6. If κ ≈ 10¹², the result is dominated by floating-point roundoff, and any answer you read off the screen is questionable past the second decimal place.

Concrete example. Take the Hilbert matrix H_n with H[i,j] = 1/(i+j-1). It's symmetric positive-definite, all the right properties on paper. But κ(H_5) is about 5×10⁵, and κ(H_10) is around 10¹³. Solve a Hilbert system Ax = b in float64 with n=10 and the answer disagrees with the true solution in the leading digits. The matrix isn't broken. Float64 just doesn't have enough resolution to represent it.

Practical heuristic: check κ(A) before computing A⁻¹b or running any iterative solver. NumPy's np.linalg.cond does it in O(n³) via the SVD. If κ exceeds 10⁸ for a problem you care about, the right move is to switch to a regularized approach. QR with column pivoting handles rank-deficiency cleanly. The Moore-Penrose pseudoinverse via SVD is the heavier option that absorbs both rank-deficiency and ill-conditioning at once. Neither makes a singular matrix invertible. They let you compute a meaningful answer instead of one whose digits are noise.

A solution exists if b lies in the column space of A—that is, b can be written as a linear combination of A's columns. Equivalently, rank([A | b]) = rank(A). If not, the system is inconsistent: no x satisfies the equation.

Uniqueness depends on the null space. If rank(A) equals the number of unknowns, the null space is trivial (only zero) and the solution is unique. If rank < unknowns, free variables exist and infinitely many solutions satisfy Ax = b.

For square invertible A, x = A⁻¹b gives the unique solution directly. In practice, solve via LU or QR decomposition rather than computing the explicit inverse—decompositions are faster and numerically stabler.

LU decomposition factors A = LU where L is lower triangular (ones on diagonal) and U is upper triangular. Solve Ax = b by first solving Ly = b (forward substitution), then Ux = y (back substitution). Faster than computing A⁻¹ explicitly.

QR decomposition factors A = QR where Q is orthogonal (Qᵀ = Q⁻¹) and R is upper triangular. More numerically stable than LU for ill-conditioned matrices. QR underlies least-squares solutions when A is rectangular (m > n).

Singular Value Decomposition (SVD) handles any matrix: A = UΣVᵀ. The singular values (diagonal of Σ) reveal rank and conditioning. SVD enables pseudoinverse computation, image compression, and principal component analysis. Computationally expensive but maximally informative.