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Unformatted text preview: GLOSSARY: A DICTIONARY FOR LINEAR ALGEBRA Adjacency matrix of a graph . Square matrix with a ij = 1 when there is an edge from node i to node j ; otherwise a ij = 0. A = A T for an undirected graph. Afne transformation T ( v ) = A v + v = linear transformation plus shift. Associative Law ( AB ) C = A ( BC ). Parentheses can be removed to leave ABC . Augmented matrix [ A b ]. A x = b is solvable when b is in the column space of A ; then [ A b ] has the same rank as A . Elimination on [ A b ] keeps equations correct. Back substitution . Upper triangular systems are solved in reverse order x n to x 1 . Basis for V . Independent vectors v 1 ,..., v d whose linear combinations give every v in V . A vector space has many bases! Big formula for n by n determinants . Det( A ) is a sum of n ! terms, one term for each permutation P of the columns. That term is the product a 1 a n down the diagonal of the reordered matrix, times det( P ) = 1. Block matrix . A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication of AB is allowed if the block shapes permit (the columns of A and rows of B must be in matching blocks). Cayley-Hamilton Theorem . p ( ) = det( A-I ) has p ( A ) = zero matrix . Change of basis matrix M . The old basis vectors v j are combinations m ij w i of the new basis vectors. The coordinates of c 1 v 1 + + c n v n = d 1 w 1 + + d n w n are related by d = M c . (For n = 2 set v 1 = m 11 w 1 + m 21 w 2 , v 2 = m 12 w 1 + m 22 w 2 .) Characteristic equation det( A-I ) = 0. The n roots are the eigenvalues of A . Cholesky factorization A = CC T = ( L D )( L D ) T for positive denite A . Circulant matrix C . Constant diagonals wrap around as in cyclic shift S . Every circulant is c I + c 1 S + + c n-1 S n-1 . C x = convolution c * x . Eigenvectors in F . Cofactor C ij . Remove row i and column j ; multiply the determinant by (-1) i + j . Column picture of A x = b . The vector b becomes a combination of the columns of A . The system is solvable only when b is in the column space C ( A ). Column space C ( A ) = space of all combinations of the columns of A . Commuting matrices AB = BA . If diagonalizable, they share n eigenvectors. Companion matrix . Put c 1 ,...,c n in row n and put n-1 1s along diagonal 1. Then det( A-I ) = ( c 1 + c 2 + c 3 2 + ). Complete solution x = x p + x n to A x = b . (Particular x p ) + ( x n in nullspace). Complex conjugate z = a-ib for any complex number z = a + ib . Then z z = | z | 2 . 1 2 Glossary Condition number cond ( A ) = ( A ) = k A kk A-1 k = max / min . In A x = b , the rela-tive change k x k / k x k is less than cond ( A ) times the relative change k b k / k b k ....
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