316 Index Schmidt, 124 Schur, 245 Schur complement, 10 Schur, trace inequality, 80 Schur, unitary triangularization theorem, 78 SDR, 174, 176 Segre characteristic, 61 sequences square summable, 25 similar matrices, 7 similarity, 7, 51 simple graph, 222 simultaneous row and column operations, 102–105 simultaneous triangularization, 151–158 Singleton bound, 271 singular value decomposition, 108–109, 121–126 singular values, 108–109, 121–122 skew-Hermitian matrix, 8 skew-symmetric matrix, 8 span, 2 spanning set, 2 Specht’s theorem, 86 spectral norm, 123 spectral radius, 46, 123, 249, 252 spectral theorem, 79, 83 spectrum, 41 spectrum of a graph, 226 sphere-packing bound, 272 square root of positive semi-definite matrix, 106 square summable sequences, 25 stable equilibrium point, 289, 291 standard inner product, 18 state vector, 279 Steiner system, 185 stochastic matrix, 295 strictly lower triangular matrix, 9 strictly upper triangular matrices, 155 strictly upper triangular matrix, 9 strong components of a digraph, 236 strongly connected, 236, 238 strongly connected vertices, 236 strongly regular graph, 227–231 subdigraph, 236 submatrix, 9 principal, 9 subspace, 2 direct sum, 2 invariant, 10 sum, 2 subword, 274 sum direct, 2 subspace, 2 sum norm, 114 supply demand model, 284 surjective, 4 Sylvester’s law of inertia, 93–94 Sylvester’s theorem, 53–54 symmetric 2-design, 190–191 symmetric bilinear form, 89 symmetric block design, 190–191 symmetric inner product, 17 symmetric matrix, 8, 139 syndrome, 273 system of distinct representatives, 174, 176 t-designs, 185–189 tensor product of Hadamard matrices, 208 tensor product of matrices, 13 tensor products, 13 term rank, 173, 177 terminal vertex, 235 T (m), 230 trace, 15, 46 trail, 224 directed, 236 transient state, 297 transition matrix, 295 transition probabilities, 295 transpose, 8 triangle in a graph, 224 triangle inequality, 113 triangular graph, 230 triangular matrix, 9, 40 triangularization, simultaneous, 151–158 tridiagonal matrix, 131–133 unitarily invariant norm, 118 unitary equivalence, 77 unitary invariants, 86 unitary matrix, 15, 32 unitary similarity, 77, 84–86 unitary space, 18 unitary transformation, 31 upper bidiagonal, 130 upper triangular matrix, 9
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