ENTRY LINEAR ALGEBRA ENTRY LINEAR ALGEBRA Author Oliver Knill Spring 2002 Spring 2004 Literature Standard glossary of multivariable calculus course as taught at the Harvard mathematics department adjacency matrix The adjacency matrix of a graph is a matrix Aij where Aij 1 whenever there is an edge from node i to node j in the graph Otherwise Aij 0 Example the graph with three nodes with the shape of a V has the 0 1 0 adjacency matrix A 1 0 1 where node 2 is conneced to both node 1 and 3 and node 1 and 3 are not 0 1 0 connected to each other affine transformation An affine transformation is the composition of a linear transformation with a shift like for example T x y 2x y 3x 4y 2 3 Algebra Algebra was originally the art of solving equations and systems of equations The word comves from the Arabic al jabr meaning restauration The term was used by Mohammed al Khowarizmi who worked in Bhagdad algebraic multiplicity The algebraic multiplicity of a root y of a polynomial p is the maximal integer k for which p x x y k q x The algebraic multiplicity is bigger or equal than the geometric multiplicity angle The angle between two vectors v and w is arccos x y x y where xy is the dot product between x and y and x x x is the length of x The inverse of cos gives two angles in 0 2 One usually choses the smaller angle argument The argument of a complex number z x iy is if z re i The argument is determined only up to addition of 2 It can be determined as arctan y x A where A 0 if x 0 or x 0 y 0 and A if x 0 or x 0 and y 0 For example arg i 2 and arg i 3pi 2 The argument is the imaginary part of log z because log reiphi log r i associative law The associative law is AB C A BC It is an identity which some mathematical operations satisfy For example the matrix multiplication satisfies the associative law One says also that the operation is associative An example of a product which is not associative is the cross product v w if i j k are the standard basis vectors then i i j i k j and i i j 0 j 0 augmented matrix The augmented matrix of a linear equation Ax b is the n n 1 matrix A b One considers the augmented matrix when solving a linear system Ax b The reduced row echelon form rref A b contains the solution vector x in the last column if a solution exists More generally a matrix which contains a given matrix as a submatrix is called an augmented matrix basis A basis of a linear space is a finite set of vectors v 1 vn which are linearly independent and which span the linear space If the basis contains n vectors the vector space has dimension n basis theorem The basis theorem states that d linearly independent vectors in a vector space of dimension d forms a basis block matrix A block matrix is a matrix A where the only non zero elements are contained in a sequence of smaller square matrices arranged along the main diagonal of A Such matrices are also called block diagonal matrices The 1 2 0 0 0 3 2 0 0 0 matrix A 0 0 5 0 0 is an example of a block diagonal matrix containing a two 2x2 and a 1x1 0 0 0 6 7 0 0 0 8 9 matrix in its diagonal Cauchy Schwarz inequality The Cauchy Schwarz inequality tells that x y is smaller or equal to x y Equality holds if and only if x and y are parallel vectors Cayley Hamilton theorem The Cayley Hamilton theorem assures that every square matrix A satisfies p A 0 where p x det A x is the characteristic polynomial of A and the right hand side 0 is the zero matrix change of basis A change of basis from an old basis vj to a new basis P wj is described by an invertible matrix S which relates the coordinates aP 1 an of a vector a i ai v i b w in the new w basis The in the old v basis with the coordinates b1 bn of the same vector b i i i P T wj where S T is the transpose of S relation of the coordinates is b Sa In that case one has v j j Sij For example if v1 1 0 v2 0 1 w1 3 4 w2 2 3 then a a1 a2 5 7 in the v basis has the 3 4 3 2 we have b Sa and and S T coordinates b b1 b2 1 1 in the w basis With S 2 3 4 3 w1 3v1 4v2 w2 2v1 3v2 characteristic matrix The characteristic matrix of a square matrix A is the matrix A x xI A where I is the identity matrix The characteristic matrix is a function of the free variable x characteristic polynomial The characteristic polynomial of a matrix A is the polynomial p x det xI A where I is the identity matrix It has the form p x xn tr A x n 1 1 n det A where tr A is the trace of A and det A is the determinant of the matrix A The eigenvalues of A are the roots of the characteristic polynomial of A Cholesky factoriztion The Cholesky factoriztion of a symmetric and positive definite matrix A is A R T R where R is upper triangular with positive diagonal entries circulant matrix A circulant matrix is a square matrix where the entries in each diagonal are constant If S is the shift 0 1 0 matrix which has 1 in the side diagonal and 0 everywhere else like in the 3x3 case S 0 0 1 then a 1 0 0 n 1 circular matrix can be written as A a a S a S A general 3x3 circulant matrix has the form 0 1 n 1 a b c A a bS cS 2 which is S c a b b c a classical adjoint The classical adjoint adj A of a n n matrix A is the n n matrix whose entry a ij is aij 1 i j det Aji where Aji is a minor of A The classical adjoint plays a role in Cramer s rule A 1 adj A det A The name adjoint comes from the fact that we have a change indices like in the adjoint However the classical adjoint has nothing to do with the adjoint codomain The codomain of a linear transformation T X Y is the target space Y The name has its origin from naming X the domain of A cofactor A cofactor Cij of a n n matrix A is the determinant of the n 1 n 1 …
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