Linear Algebra

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25 domain-specific abstractions whose origin domain is Linear Algebra.

• Bidiagonal matrix — A banded matrix whose potentially nonzero entries lie only on the main diagonal and one adjacent superdiagonal or subdiagonal, yielding simple determinants, eigenvalues, products, and efficient structured computations.
• Cauchy matrix — A structured matrix with entries 1/(x_i−y_j) for distinct parameter sequences with nonzero cross-differences, possessing explicit determinant, inverse, displacement rank, and totally nonsingular submatrix formulas.
• Circulant matrix — A square matrix generated by cyclically shifting one row, equivalently with entries depending only on index difference modulo n, and diagonalized by the discrete Fourier transform.
• Corank — A rank-deficiency quantity, commonly the codomain dimension minus rank of a linear map or matrix, equivalently the dimension of its cokernel in finite-dimensional linear algebra.
• Defective matrix — A square matrix lacking a full basis of eigenvectors and therefore not diagonalizable over the stated field.
• Definite quadratic form — A real quadratic form that is strictly positive on every nonzero vector or strictly negative on every nonzero vector.
• Dimension (vector space) — The cardinality of any basis of a vector space over a specified field, well-defined because all bases have the same cardinality.
• Dot Product — A symmetric bilinear pairing of real coordinate vectors that sums componentwise products and thereby encodes Euclidean length, angle, orthogonality, and projection.
• Exchange matrix — The permutation matrix with ones on the antidiagonal that reverses coordinate, row, or column order.
• Frobenius normal form — Replace a square matrix over a field by the unique block diagonal matrix of companion matrices determined by its divisibility-ordered invariant factors, thereby deciding similarity without splitting the characteristic polynomial.
• Hadamard product (matrices) — The entrywise product of two matrices of identical shape, multiplying corresponding entries without summing across indices.
• Linear complex structure — A real-linear endomorphism J of a real vector space satisfying J squared equals minus the identity, thereby defining multiplication by complex scalars.
• Linear map — A function between vector spaces that preserves vector addition and scalar multiplication, equivalently preserving every finite linear combination.
• M-matrix — A real Z-matrix expressible as a nonnegative scalar multiple of the identity minus a nonnegative matrix with scalar at least its spectral radius.
• Matrix congruence — An equivalence relation on square matrices in which B equals transpose-P times A times P for an invertible change-of-basis matrix P.
• Minor (linear algebra) — The determinant of a square submatrix obtained by selecting equal-size subsets of a matrix’s rows and columns.
• Modal matrix — A matrix whose columns are eigenvectors of a square matrix, used as the change of basis that diagonalizes it when a full eigenbasis exists.
• Outer product — Map two coordinate vectors to the rank-at-most-one matrix whose ij entry is the product of the first vector’s i component and the second vector’s j component.
• Scalar multiplication — The vector-space or module operation that combines a scalar with a vector to produce another vector.
• Semilinear map — An additive map between vector spaces whose scalar multiplication is respected after applying a fixed field automorphism.
• Sesquilinear form — A two-argument form on complex vector spaces that is linear in one argument and conjugate-linear in the other.
• Transpose of a linear map — The induced linear map between dual spaces obtained by precomposing functionals with the original map.
• Unimodular matrix — A square integer matrix with determinant plus or minus one, equivalently an integer matrix invertible over the integers.
• Weyr canonical form — A canonical matrix form obtained by regrouping Jordan chains by level, yielding a block structure particularly suited to describing matrices that commute with a given operator.
• Z-matrix (mathematics) — A real square matrix whose every off-diagonal entry is nonpositive.