Linear algebra

5 articles

Matrix Determinant System of linear equations Vector space Linear transformation Eigenvalues and eigenvectors Inner product space Orthogonality Diagonalization Singular value decomposition Absolutely convex set Antilinear map Antiunitary operator Asymmetric norm Basis function Basis linear algebra Basis vector Bra ket notation Canonical basis Cauchy schwarz inequality Change of basis Controlled invariant subspace Convex cone Coordinate vector Covector Cramers rule Cyclic subspace Cyclic vector Delta operator Dimension Dimension theorem for vector spaces Dimension vector space Dot product Dual basis Dual norm Dual space Euclidean norm Euclidean vector Euclidean vector space Examples of vector spaces Flag linear algebra Frame linear algebra Fredholm alternative Fredholms theorem Grobner basis Hamel basis Hamel dimension Homogeneous linear equation Hyperplane Independent equation Inner product Invariant convex cone K frame Kernel linear algebra Linear combination Linear dependence Linear form Linear function Linear functional Linear independence Linear inequality Linear map Linear span Linear subspace Norm Normal basis Normed vector space Nullspace property Orientation of a vector bundle Orientation vector space Orthant Orthogonal basis Orthogonal complement Orthogonal procrustes problem Orthogonal projection Orthogonal transformation Orthogonalization Orthonormal basis Orthonormal function system Orthonormality Overcompleteness Overdetermined system Parallel operator Partial trace Predual Prehomogeneous vector space Projection linear algebra Projectivization Quasinorm Quotient space linear algebra Radial set Reducing subspace Rotas basis conjecture Scalar Scalar multiplication Semilinear map Seminorm Spherical basis Standard basis Sublinear function Symmetric set Symplectic vector space Total set Underdetermined system Unit vector Unitary transformation Vector mathematics and physics Zero vector Absolute neighbourhood extensor Affine tensor Anti eigenvalue Banach jordan algebra Banach jordan pair Bilinear differential Bilinear functional Bilinear integral form Bilinear mapping Bimatrix game Bivector space Bochner curvature tensor Complete problem of eigen values Complexification Contraction of a tensor Contravariant tensor Covariant tensor Curvature tensor Darboux tensor Deformation tensor Degenerate matrix Determinant variety Deviatoric tensor Dirac matrices Discriminant tensor Eigen function Eigen oscillation Eigen value Eigen values of differential operators numerical methods Eigen values of integral operators numerical methods Eigen vector Equivalent matrix Fractional linear mapping Fredholm eigenvalue of a jordan curve Frobenius matrix norm Gram determinant Information matrix Jacobi matrix Jordan criterion Jordan decomposition Jordan dedekind lattice Jordan dedekind property Jordan dedekind space Jordan k roly Jordan lemma Jordan measure Jordan theorem Jordan totient function Jordan triple system Knots and links quadratic forms of Matrix element Matrix factorization method Matrix game Matrix of transition probabilities Matrix summation method Matrix tree theorem Matrix variate elliptically contoured distribution Matrix vi te theorem Measure in a topological vector space Minimal polynomial Monomial matrix Moore determinant Multilinear mapping Neumann eigenvalue Non derogatory matrix Non singular matrix Nuclear bilinear form Oscillating matrix Partial problem of eigen values Partially specified matrices completion of Pseudo quadratic form Pseudo tensor Quadratic forms reduction of Quaternary quadratic form Regression matrix Ricci tensor Riemann christoffel tensor Riemann tensor Schur determinant lemma Schur stability of polynomials and matrices Semi linear mapping Semi simple matrix Similar matrix Skew symmetric bilinear form Skew symmetric tensor Spherical matrix distribution Stress tensor Submatrix Symmetric tensor Tensor on a vector space Tits quadratic form Transposed matrix Vandermonde determinant Williamson matrices Eigenspaces

Probability & Statistics LicenceExplication9 min

Twenty-Four Per Cent Minus Twice the Floating Rate Is Three Ordinary Swaps

A pays the floating rate L and receives 24 per cent minus 2L, which nets to 24 minus 3L and factors as three times 8 minus L: three vanilla swaps at eight per cent, so the fixed rate was never the twenty-four printed on the deal. Reading it as twenty-four is a 48-point error at a floating rate of twenty-four. The article also carries the version that does need a model, where a floor on the inverse leg adds two caplets struck at twelve and the factorisation fails.

Probability & Statistics LicenceExplication10 min

Slide It, Stretch It, and the Correlation Does Not Move

A shift leaves every deviation from the mean untouched, and a stretch multiplies the covariance and one standard deviation by the same factor, so both cancel out of the ratio. The tempting answer of five times rho is worse than wrong: at rho = 0.40 it names 2.0, which Cauchy-Schwarz forbids any correlation from reaching. The article carries the general affine rule, the sign flip a negative factor produces, and the curved maps the invariance does not survive.

Probability & Statistics LicenceAnalyse14 min

Two Stocks at 0.7 With the Same Index, and Still −0.02 With Each Other

The two correlations you are handed do not pin the third one down, but they fence it into exactly [−1/50, 1], and that interval dips below zero. The fence falls out of a 3×3 determinant read as a quadratic in the unknown, and out of a picture: 0.7 is an angle of 45.573°, both stocks live on a cone of that half-angle around the index, and putting them on opposite sides opens 91.146° between them. Also here: why the real tipping point is ab ≥ 0 together with a² + b² ≥ 1 rather than "both above 0.707", why 0.9 and 0.5 force a positive answer while 0.9 and −0.9 allow −1, why standing on the floor costs a rank, and why three Bernoulli(0.5) indicators with the same two correlations are confined to [0.40, 1] instead.

Combinatorics & Discrete Math LycéeAnalyse10 min

Nine Digits, Four Windows, and a Middle That Cannot Move

Nine slots hold the digits 1 to 9 and four overlapping windows each add to 20. One subtraction pins the centre digit to 5 with no search at all, and the same structure classifies every arrangement: there are 96 of them, falling into 48 mirror pairs. Change the target and the centre is forced to an odd digit that sometimes admits nothing.

Geometry LicenceAnalyse11 min

A Mirror Reverses One Axis, and It Is Not Left or Right

Raise your right hand at a mirror and the reflected hand stays on the same side of the room, which means the usual question has a false premise. A plane mirror is the matrix diag(1, 1, -1): it fixes both axes lying in the glass and reverses only the direction you look along. Its determinant is -1, so no rotation reproduces it, and the sideways flip everyone reports belongs to the half turn you perform in your head.