linalg4 http://www.openmath.org/cd/linalg4.ocd 2003-04-01 1999-07-15 2 0 experimental alg1 arith1 linalg1 linalg2 linalg5 logic1 relation1 This CD defines symbols for basic linear algebra. Regardless of the way of forming vectors and matrices, this CD deals with eigenvalues, eigenvectors and related concepts. eigenvalue This symbol represents the eigenvalue of a matrix. It takes two arguments the first should be the matrix, the second should be an index to specify the eigenvalue. The ordering imposed on the eigenvalues is first on the modulus of the value, and second on the argument of the value. A definition of eigenvalue is given in Elementary Linear Algebra, Stanley I. Grossman in Definition 1 of chapter 6, page 533. A*eigenvector(A,i) = eigenvalue(A,i)*eigenvector(A,i) eigenvector This symbol represents the eigenvector of a matrix. It takes two arguments the first should be the matrix, the second should be an index to specify which eigenvalue this eigenvector should be paired with. The ordering is as given in the eigenvalue symbol. A definition of eigenvector is given in Elementary Linear Algebra, Stanley I. Grossman in Definition 1 of chapter 6, page 533. A*eigenvector(A) = eigenvalue(A)*eigenvector(A) characteristic_eqn This symbol represents the polynomial which appears in the left hand side of the characteristic equation of a matrix. It takes one argument which should be the matrix. A definition of the characteristic equation is given in Elementary Linear Algebra, Stanley I. Grossman in Definition 2 of chapter 6, page 535. p(eigenvalue(A,i)) = det(A-eigenvalue(A,i)I) = 0 where p is the characteristic equation of A size This symbol represents the function which takes one vector argument and returns the length of that vector. the length of the vector [1,2,3] = 3 1 2 3 3 rank This symbol represents the function which takes one matrix argument and returns the number of linearly independent rows (or columns) of that matrix. the rank of an nxn identity matrix is n rowcount This symbol represents the function which takes one matrix argument and returns the number of rows in that matrix. Specification of the number of rows in the matrix: [[1 2] [3 4] [5 6]] 1 2 3 4 5 6 columncount This symbol represents the function which takes one matrix argument and returns the number of columns in that matrix. Specification of the number of columns in the matrix: [[1 2] [3 4] [5 6]] 1 2 3 4 5 6