Compute Core Matrix Properties Up to 4x4

Linear algebra underpins everything from machine learning models to circuit analysis, translating systems of equations into matrix form for systematic solution. A data scientist building a recommendation system needed to check if a user-item matrix had full rank before applying collaborative filtering. She computed rank and found it deficient—some items were linear combinations of others—explaining why her model kept producing degenerate results. The common mistake is assuming every square matrix is invertible. When interpreting results, always verify that rank equals the matrix dimension before trusting inverse computations, and remember that eigenvalues encode stability and principal directions.

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.

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.