408 INDEX
quadratic form, 168quotient space, 212quotient vector space, 212
range, 10rank, 117
number of pivot columns, 120rank of a matrix, 99, 116rank one transformation, 281rational canonical form, 255
uniqueness, 258Rayleigh quotient, 368
how close?, 368real numbers, 10real Schur form, 162regression line, 283regular Sturm Liouville problem, 286relatively prime, 22Riesz representation theorem, 278right Cauchy Green strain tensor, 308right polar decomposition, 309row equivalelance
determination, 120row equivalent, 119row operations, 27, 111
inverse, 28linear relations between columns, 117
row rank, 99, 116row reduced echelon form
definition, 118examples, 119existence, 119uniqueness, 120
scalar product, 33scalars, 16, 32, 37Schur’s theorem, 160, 298
inner product space, 298second derivative test, 172self adjoint, 164, 279self adjoint nonnegative
roots, 307set notation, 9sgn, 89
uniqueness, 91shifted inverse power method, 361
complex eigenvalues, 366sign of a permutation, 91similar
matrix and its transpose, 254similar matrices, 88, 108, 219
similarity transformation, 219simple field extension, 207simultaneous corrections, 342simultaneously diagonalizable, 296
commuting family, 298singular value decomposition, 313singular values, 313skew symmetric, 47, 156space of linear transformations
vector space, 281span, 60, 94spanning set
restricting to a basis, 194spectral mapping theorem, 400spectral norm, 329spectral radius, 335, 336spectrum, 143splitting field, 203stochastic matrix, 263subspace, 60, 190
basis, 64, 195complementary, 292dimension, 64invariant, 237
subspacesdirect sum, 234direct sum, basis, 235
substituting matrix into polynomial identity,110
surjective, 10Sylvester, 80
law of inertia, 183dimention of kernel of product, 233
Sylvester’s equation, 292symmetric, 47, 156system of linear equations, 29
tensor product, 281the space AU, 292trace, 167
AB and BA, 167sum of eigenvalues, 178
transpose, 47properties, 47
triangle inequality, 34trivial, 60
union, 9Unitary matrix
representation, 356upper Hessenberg matrix, 383