Index 315 field of values, 144 n-set, 171 null space, 3 nullity, 4 number of walks of length m in a graph, 225 numerical range, 143 one-to-one, 4 onto, 4 open walk, 224 operator norm, 123 order of a graph, 221 order of a projective plane, 191, 195 orthogonal complement, 23 projection onto a line, 19–20 projection onto a subspace, 23–26 set, 19 vectors, 19 orthogonal matrix, 32 orthogonal projection onto a line, 19–20 onto a subspace, 23–26 orthonormal, 19 orthonormal basis, 21 orthostochastic, 178 outdegree, 235 outer product, 18 Paley, 208 Paley’s theorem, 209 parity check, 266 parity check matrix, 267–268, 271 Parker, 81 partitioned conformally, 10 partitioned matrices, 9 Pearcy, 86 pencil, 158 perfect code, 272 permanent, 175–176 permutation matrix, 172 Perron’s theorem, 249, 254–257 Perron–Frobenius theorem, 249 Petersen graph, 227 plane, 3 polar decomposition, 107–108, 126–127 polar factorization, 107–108 population cohort model, 281 positive definite matrix, 22, 99–102, 139 eigenvalues, 100 positive matrix, 249 positive semi-definite, 99 positive semi-definite matrix square root of, 106 positivity, 17 power method, 135 power series, 288 primitive graph, 242 primitive matrix, 245, 260–262 principal submatrix, 9, 47 principal submatrix of a Hermitian matrix, 97 probability vector, 295 product of matrices, 4 projection, 15, 49 projective plane of order 2, 185, 186, 191 projective planes, 191–197 proper values, 40 property L, 158–161 property P, 152 Pythagorean theorem, 18–19 QR factorization, 33, 129–130 quadratic character, 209 quadratic form, 198–199 quadratic residue matrix, 209–211 range space, 4 rank, 4, 47 column, 5 of a linear transformation, 4 of a matrix, 5 row, 5 rank k approximation, 123–125 rank plus nullity theorem, 4 rational canonical form, 64 Rayleigh quotient, 94 Rayleigh–Ritz ratio, 94 reducible matrix, 238 regular graph, 223 degree of, 223 regular of degree k, 236 repetition code, 265–266 right shift operator, 31 row echelon form, 138 row operations, 102–105 row rank, 5 row space, 5 row stochastic matrix, 295 scalar, 2 Scarpis, 208

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