附录E Weierstrass定理

设 $V$ 是具有范数 $\|\cdot\|$ 的有限维实或复向量空间。关于中心为 $x \in V$ ，半径为 $\varepsilon$ 的球是 $B_{\varepsilon}(x) = \{y \in V : \|y - x\| \leqslant \varepsilon\}$ 。我们称子集 $S \subseteq V$ 是开集，是指对每个 $x \in S$ ，都存在一个 $\varepsilon > 0$ ，使得 $B_{\varepsilon}(x) \subseteq S$ 。子集 $T \subseteq V$ 称闭集，是指 $T$ 在 $V$ 中的补集是开集。子集 $S \subseteq V$ 称为有界的，是指存在 $r > 0$ ，使得 $S \subseteq B_r(0)$ 。等价地， $T$ 是闭的，当且仅当 $T$ 的任一收敛序列（关于 $\|\cdot\|$ ）的极限都在 $T$ 中，而 $S$ 是有界的，是指 $S$ 包含在具有有限半径的任一球中。子集 $S \subseteq V$ 是紧集，是指它既是闭的，又是有界的。

对于 $S \subseteq V$ ，一个函数 $f: S \to \mathbb{R}$ 在 $S$ 上可以达到或者不可以达到一个（全局）极大值或极小值。但是，在某些常见的情形，我们可以确信， $f$ 在 $S$ 上达到一个极大值。

定理（Weierstrass）设 $S$ 是有限维实或复向量空间 $V$ 的紧子集。如果 $f: S \to \mathbb{R}$ 是连续函数，那么存在点 $x_{\min} \in S$ ，使得对所有 $x \in S$ 有

且存在点 $x_{\max} \in S$ ，使得对所有 $x \in S$ 有

即 $f$ 在 $S$ 上达到它的极小值和极大值。当然，可能不止在 $S$ 的一个点上达到值 $\max_{x \in S} f(x)$ 和 $\min_{x \in S} f(x)$ ，如果 Weierstrass 定理的两个主要假定（紧的 $S$ 和连续的 $f$ ）不成立，结论可能不真。但是， $S$ 是一个有限维实或复向量空间的子集不是本质的。对子紧集的一个适当的定义，Weierstrass 定理对定义在一般拓扑空间的一个紧子集上的连续实值函数成立。

参考文献

Ait A. C. Aitken. Determinants and Matrices. 9th ed. Oliver and Boyd, Edinburgh, 1956.
Bar 75 S. Barnett. Introduction to Mathematical Control Theory. Clarendon Press, Oxford, 1975.
Bar 79 S. Barnett. Matrix Methods for Engineers and Scientists. McGraw-Hill, London, 1979.
Bar 83 S. Barnett. Polynomials and Linear Control Systems. Dekker, New York, 1983.
BB E. F. Beckenbach and R. Beliman. Inequalities. Springer-Verlag, New York, 1965.
Bel R. Bellman. Introduction to Matrix Analysis. 2d ed. McGraw-Hill, New York, 1970.
Boa R. P. Boas, Jr. A Primer of Real Functions. 2d ed. Carus Mathematical Monographs, No. 13. Mathematical Association of America, Washington, D.C., 1972.
BPl A. Berman and R. Plemmons. Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York, 1979.
BSt S. Barnett and C. Storey. Matrix Methods in Stability Theory. Barnes & Noble, New York, 1970.
CaLe J. A. Carpenter and R. A. Lewis. KWIC Index for Numerical Algebra. U.S. Dept. of Commerce, Springfield, Va. Microfiche and printed versions available from National Technical Information Service, U.S. Dept. of Commerce, 5285 Port Royal Road, Springfield, VA 22161.
Chi L. Childs. A Concrete Introduction to Higher Algebra. Springer-Verlag, Berlin, 1979.
Cul C. G. Cullen. Matrices and Linear Transformations. Addison-Wesley, Reading, Mass., 1966.
Don W. F. Donoghue, Jr. Monotone Matrix Functions and Analytic Continuation. Springer-Verlag, Berlin, 1974.
Fad V. N. Faddeeva. Trans. C. D. Benster. Computational Methods of Linear Algebra. Dover, New York, 1959.
Fan Ky Fan. Convex Sets and Their Applications. Lecture Notes, Applied Mathematics Division, Argonne National Laboratory, Summer 1959.
Fie M. Fieldler. Spectral Properties of Some Classes of Matrices. Lecture Notes, Report No. 75.01R. Chalmers University of Technology and the University of Göteborg, 1975.
Fra J. Franklin. Matrix Theory. Prentice-Hall, Englewood Cliffs, N.J., 1968.
Gan F. R. Gantmacher. The Theory of Matrices. 2 vols. Chelsea, New York, 1959.
Gant F. R. Gantmacher. Applications of the Theory of Matrices. Interscience, New York, 1959.

GKr F. R. Gantmacher and M. G. Krein. Oszillationsmatrizen, Oszillationskerne, und keine Schwingungen mechanische Systeme. Akademie-Verlag, Berlin, 1960.
GLR 82 I. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, New York, 1982.
GLR83 I. Gohberg, P. Lancaster, and L. Rodman. Matrices and Indefinite Scalar Products. Birkhäuser-Verlag, Boston, 1983.
Grah A. Graham. Kronecker Products and Matrix Calculus with Applications. Horwood, Chichester, U.K., 1981.
Gray F. A. Graybill. Matrices with Applications to Statistics. 2d ed. Wadsworth, Belmont, Calif., 1983.
Gre W. H. Greub. Multilinear Algebra. 2d ed. Springer-Verlag, New York, 1978.
GVI G. Golub and C. VanLoan. Matrix Computations. Johns Hopkins University Press, Baltimore, 1983.
Hal58 P. R. Halmos. Finite-Dimensional Vector Spaces. Van Nostrand, Princeton, N.J., 1958.
Hal67 P. R. Halmos. A Hilbert Space Problem Book. Van Nostrand, Princeton, N.J., 1967.
HJ R. Horn and C. Johnson. Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1989.
HKu K. Hoffman and R. Kunze. Linear Algebra. 2d ed. Prentice-Hall, Englewood Cliffs, N.J., 1971.
Hou 64 A. S. Householder. The Theory of Matrices in Numerical Analysis. Blaisdell, New York, 1964.
Hou 72 A. S. Householder. Lectures on Numerical Algebra. Mathematical Association of America, Buffalo, N.Y., 1972.
HSm M. W. Hirsch and S. Smale. Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, New York, 1974.
Jac N. Jacobson. The Theory of Rings. American Mathematical Society, New York, 1943.
Kap 1. Kaplansky. Linear Algebra and Geometry: A Second Course. Allyn & Bacon, Boston, 1969.
Kar S. Karlin. Total Positivity. Stanford University Press, Stanford, Calif., 1960.
Kel R. B. Kellogg. Topics in Matrix Theory. Lecture Notes, Report No. 71.04, Chalmers Institute of Technology and the University of Goteberg, 1971.
Kow H. Kowalsky. Lineare Algebra. 4th ed. deGruyter, Berlin, 1969.
LaH C. Lawson and R. Hanson. Solving Least Squares Problems. Prentice-Hall, Englewood Cliffs, N.J., 1974.
P.Lancaster.TheoryofMatrices.AcademicPress,NewYork,1969.
LaTi P. Lancaster and M. Tismenetsky. The Theory of Matrices With Applications. 2d ed. Academic Press, New York, 1985.
Mac C.C.MacDuffee.The Theory ofMatrices.Chelsea,New York,1946.
Mar M. Marcus. Finite Dimensional Multilinear Algebra. 2 vols. Dekker, New York, 1973-75.
Air L. Mirsky. An Introduction to Linear Algebra. Clarendon Press, Oxford, 1963.
MMi M. Marcus and H. Minc. A Survey of Matrix Theory and Matrix Inequalities. Allyn & Bacon, Boston, 1964.
MOI A. W. Marshall and I. Olkin. Inequalities: Theory of Majorization and Its Applications. Academic Press, New York, 1979.

Mui T. Muir. The Theory of Determinants in the Historical Order of Development. 4 vols. MacMillan, London, 1906, 1911, 1920, 1923; Dover, New York, 1966. Contributions to the History of Determinants, 1900-1920. Blackie, London, 1930.
Ner E. Nering. Linear Algebra and Matrix Theory. 2d ed. Wiley, New York, 1963.
New M.Newman. Integral Matrices.Academic Press, New York, 1972. Nob B.Noble.Applied Linear Algebra.Prentice-Hall, Englewood Clift N.J.,1969.
Per S.Perlis.Theory of Matrices.Addison-Wesley,Reading,Mass.,1952.
Rog G. S. Rogers. Matrix Derivatives. Lecture Notes in Statistics, Vol. 2. Dekker, New York, 1980.
Rud W. Rudin. Principles of Mathematical Analysis. 3rd ed. McGraw-Hill, New York, 1976.
Sen E. Seneta. Nonnegative Matrices. Wiley, New York, 1973.
Ste G. W. Stewart. Introduction to Matrix Computations. Academic Press, New York, 1973.
Str G. Strang. Linear Algebra and Its Applications. Academic Press, New York, 1976.
STy D. A. Suprenenko and R. I. Tyshkevich. Commutative Matrices. Academic Press, New York, 1968.
Tod J. Todd (ed.). Survey of Numerical Analysis. McGraw-Hill, New York, 1962.
TuA H. W. Turnbull and A. C. Aitken. An Introduction to the Theory of Canonical Matrices. Blackie, London, 1932.
Tur H. W. Turnbull. The Theory of Determinants, Matrices and Invariants. Blackie, London, 1950.
Val F.A.Valentine.Convex Sets.McGraw-Hill,New York,1964.
Var R. S. Varga. Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, N.J., 1962.
Wed J. H. M. Wedderburn. Lectures on Matrices. American Mathematical Society Colloquium Publications XVII. American Mathematical Society, New York, 1934.
Wie H. Wielandt. Topics in the Analytic Theory of Matrices. Lecture Notes prepared by R. Meyer. Department of Mathematics, University of Wisconsin, Madison, 1967.
Wil J. H. Wilkinson. The Algebraic Eigenvalue Problem. Clarendon Press, Oxford, 1965.

索引

索引中的页码为英文原书页码。与书中页边标注的页码一致。

A

a priori bounds (先验界), 337

absolute (绝对)

convergene (收敛). 279, 300

value of a complex number (复数的值), 532

vector norm (向量范数), 285, 310, 365. 438

additive property, of inner product (内积的可加性), 260

adjoint (伴随)

classical (经典), 20

Hermitian (Hermite), 6

adjugate (转置伴随), 20

algebraically (代数)

closed field (闭域), 41, 537

alternating sum (交错和), 8

angle between vectors (向量间夹角), 15

annihilate (零化), 142

antilinear transformation (反线性变换), 250

approximation problems (逼近问题), 332, 427

augmented matrix (增广矩阵). 11. 12

B

back substitution (后向替换), 159

ball of radius $r$ （半径为 $r$ 的球）. 281, 541

basis (基), 3

change of (变换), 30

orthonormal (正交), 16

representation (表示), 31

bilinear form (双线性型), 169, 175

biorthogonality (双正交性), 59

Birkhoff's theorem (Birkhoff定理), 197, 527

Bochner's theorem (Bochner定理), 394

boundary (边界), 282

bounded set (有界集) 282, 541

Brauer

condition for invertibility (可逆条件), 381

region (区域), 380

theorem (定理), 380

Brualdi

condition for invertibility (可逆性条件), 389

region (区域)、385

theorem (定理), 385, 387

C

cancellation theorem (消去定理), 78, 141

canonical forms (标准形)

consimilarity (合相似). 251

integer matrices (整数矩阵). 158

irreducible normal form (不可约), 506

Jordan, 121

rational canonical (有理), 156

rational matrices (有理矩阵), 158

real Jordan (Jordan), 152

real orthogonal matrices (实正交矩阵), 108

rcal skew-symmetric matrices (实斜对称矩阵), 107

real symmetric matrices (实对称矩阵), 107

singular value decomposition (奇异值分解), 157.414ff

symmetric Jordan (对称 Jordan), 209

triangular factorization (三角分解), 157

Carmichael and Mason's bound on zeroes (关于零点的 Carmichael 界), 317, 318, 364

Cassini, ovals of (Cassini 椭圆形), 380

Cauchy

sequence (序列), 274

bound on zeros (关于零点的界), 316, 318, 364

(Cauchy Binet formula (Cauchy Binet 公式), 22)

Cayley-Hamilton theorem (Cayley-Hamilton 定理). 86

characteristic equation (特征方程), 87

characteristic polynomial (特征多项式), 38, 86, 87, 540

Cholesky factorization (Cholesky分解). 114, 407

closed set (闭集), 282, 541

closure (闭包). 282

cofactor (代数余子式). 17

column rank (列秩), 12

commutative ring (交换环). 95

commutator (换位了). 98

commuting family (交换族). 51, 81, 99, 139

compact set (紧集), 282, 541

completeness property of a vector space (向量空间的完备性), 271

complex (复)

conjugate (共轭), 531

numbers (数), 531, 532

concave function (凹函数), 534

condiagonalization (合对角化), 244, 248

condition number (条件数), 336, 340, 365, 366, 374, 442

conegenvalue （合特征值）

characterization (特征). 246

definition (定义). 245

coneigenvector (合特征向量), 245

conformal (共形的), 17

congruence （相合）

rongruence (相合), 220, 399, 164ff, 170

simultaneouscongruence,canonicalpairs（同时相合标准形偶）.236

congruence (1) 220

conjugate linear (共轭线性). 169

conjunctive (共轭相合). 220

consimilarity (合相似), 234, 244

characterizations (的特征), 251

to a real matrix (于一个实矩阵), 255

consistent (相容)

linear equations (线性方程组), 12

vector norm (向量范数), 324

constrained extrema (约束极值), 34

continuous dependence of eigenvalues (特征值的连续依赖), 540

contriangularization (合三角化), 244

convergence of a sequence (序列的收敛), 269

convex (凸), 284

combination (组合), 535

cone (锥). 463

function (函数), 392, 533, 534-536

hull (包), 533

seis (集), 533-536

coordinate representation (坐标表示), 30

Courant-Fischer theorem (Courant Fischer定理), 179, 420, 424, 472

Cramer's rule (Cramer法则), 21

cycle (同路), 357

cyclic of index $k$ （指标 $k$ 的循环），514

D

defect from normality (正规性亏损值), 316

defectiv (亏损), 58

deflation (压缩), 63, 83

delcted absolute row sums (去心绝对行和), 344

dependent (相关), 3

determinant (行列式), 7. 11. 398

determinantal inequalities (行列式不等式), 453, 167, 476-486

Fischer, 178

Gersgorin. 351

Hadamard. 477

Hadamard-Fischer, 485

Minkowski, 482

Oppenheim, 480

Ostrowski-Taussky, 481

Sasz. 479

diagonalizable (可对角化), 139, 145

hy orthogonal similarity (用正交相似), 211

definition (定义), 46

orthogonality (正交), 101

simultaneously (同时). 49

unitary (西), 101

diagonalization (对角化)

by congruence (用相合), 228

by consummolarity (用合相似), 234, 244, 248

by similarity (用相似), 46, 145

by unitary congruence (用酉相合), 204

by unitary consimilarity (用酉合相似), 244, 245

by unitary similarity (用酉相似), 101

simultaneous (同时), 52

diagonally dominant (对角占优), 349

strictly (严格), 302, 349

difference scheme (差分方法), 394

differential equations (微分方程), 132, 394

elliptic (椭圆型), 239, 459

hyperbolic (双曲型), 239

partial (偏), 168, 216, 218

dimension (维数). 4

direct sum (占和), 24

directed (有向)

graph of a matrix (矩阵图). 357, 517, 522

path (道路), 357

dual pair (对偶对), 278

duality theorem (对偶性定理), 287

E

edgs (边), 168

cigenspace (特征空间), 57

cigenvalue （特征值）

algebraic multiplicity (代数重数), 58, 60, 138.

1. 497, 499

algebraically simple (代数单重). 371

definition (定义), 35

deflation to calculate (计算的压缩), 63

distinct (互异), 48

dominant (优势), 506

generalized (广义), 213

geometric multiplicity (几何重数), 58, 60.

1. 141, 497, 498

illconditioned （病态），367

inclusion region (包含区域), 501

inclusion theorem (包含定理), 177

location (估计). 343

moments (矩). 43

of a sum (和的), 181, 184

perfectly conditioned (优态). 367

power method to calculate (求的解法), 62

principal submatrices (主子矩阵的), 189

well-conditioned (良态), 367

cigenvector (特征向量), 57

definition (定义), 35

left (左), 59, 371

positive (正), 493, 494, 495, 513

right (右), 59

elementary divisors (初等因子), 155

elementary symmetric functions (初等对称函数), 41

elliptic differential operator (椭圆型微分算子), 239

cquilibrated (均衡的), 283

equivalence relation (等价关系)

congruence (相合), 221

consimilarity (合相似), 251

definition (的定义), 45

vector seminorm (向量半范数的), 262

equivalent （等价）

matrices (矩阵), 164

orthogonally (上交), 73

real orthogonally (实正交), 73

unitarily (酉), 72

vector norms (向量范数), 273, 279

error analysis (误差分析), 335

Euler's theorem (Euler定理), 111

exponential of a matrix (矩阵指数), 300

extreme points （端点）

closed convex set (闭凸集), 533

doubly stochastic matrices (双随机矩阵), 528

extreme ray (极射线)

definition (定义), 463

positive semidfinite matrices (半正定矩阵的), 164

F

factor analysis (因子分析), 431

factorizations (分解), 156

Cholesky, 114. 407

complex skew-symmetric matrix (复斜对称矩阵), 217

complex symmetric matrix (复对称矩阵), 204

L.U. 158-165

polar (极), 156, 411, 412ff

product of two Hermitian matrices (两个 Hermite 矩阵之乘积), 172

QR, 112. 164, 406

singular value decomposition (奇异值), 411

Takagi，250，423，166

triangular 157

family （矩阵）族）

commuting (交换), 51, 81, 99, 139

commuting real normal (实正规交换), 108, 112

complex symmetric (复对称), 243

diagonalizable symmetric (可交换对称), 217

Hermitian (Hermite), 172

normal (1.规), 103

simultaneous condiagonalization (同时合对角化), 252

simultaneous diagonalization by congruence (经相合同时对角化), 239

simultaneous diagonalization by unitary congruence (经(相合同时对角化), 213

simultaneous singular value decomposition (同时奇异值分解), 126

simultaneous triangularization (同时角化), 84

Fan （樊）

k norms (k 范数), 145

theorem on eigenvalue location (关于特征值位置的定理), 501

Fejer (Fejer)

trace theorem, on positive semidefinite matrices

（关于半正定矩阵的迹定理），459

uniqueness theorem. for elliptic partial differential

equations （关于椭圆型偏微分方程的唯一性定理）, 460

field (域).

of values (值域). 321. 332

forms (型)

bilinear (双线性), 169, 175

Hermiua (Hemite), 174

quadratic (二次), 168, 174, 214, 466

sesquilinear (半双线性), 169

forward substitution (前向替换), 159

fundamental theorem of algebra (代数基本定理), 537

G

general linear group (一般线性群), 14

generalized （广义）

coordinates (坐标), 227

inverse (逆), 421

matrix functions (矩阵函数), 8

Gersgorin

circles (圆), 346

disc theorem (圆盘定理), 344

discs (圆盘), 345, 353

region (区域), 345

Givens's method (Givens 法), 77

Gram-Schmidt process (Gram-Schmidt 过程), 15. 148

modified (修改的), 116

symmetric analogue (对称矩阵的类似), 211

graph (图), 168

group （群）

finiteAbelian (有限Abel),510

general linear (一般线性). 14

isometry (等距), 266, 267

orthogonal (正交), 69, 71

unitary (西), 69

11

Hadamard

exponential of a matrix (矩阵指数). 161

inequality（不等式）.199，200，477，483

powers of a matrix (矩阵), 462

product (乘积), 321, 455, 456, 457, 474, 475

square root of a matrix (矩阵方根), 162

Hahn Banach theorem (Hahn-Banach 定理), 288

half spaces (半空间), 534

Hermittian

part (部分), 109, 170, 399

property (性质), 260

Hermiteian matrices (Hermite矩阵)

congruent (相合), 223, 224

product of three （三个Hermite 矩阵的乘积），469

product of two (两个Hermite矩阵的乘积), 172

Hermitian matrix (Hermite 矩阵), 104, 167, 169, 397

analogous to real numbers (比作实数), 170

characterizations (特征), 171

paritioned (分块), 175

product with positive definite matrix (与正定矩阵的乘积), 465

spectral theorem (的谱定理), 171

Hessian (Hessian)167,392，459，534

Hoffman-Wielandt theorem (Hoffman-Wielandt 定理). 368, 119

homogeneous (齐次性), 259, 260, 290

Hopf's bound (Hopf 界), 501

Householder

transformation (变换), 74, 77, 78, 117

method (法), 78

hyperbolic differential operator (双曲型微分算子), 239

hyperplane (超平面), 534

1

idempotent (解等), 37, 148, 311

identity (恒等式)

Newton. 11

parallelogram (平行四边形), 263

polarization (极化), 263

Sylvester. 22

ill-conditioned (病态), 336

imaginary (座)

axin (虚轴), 532

part of a complex number (复数的虚部), 531

inclusion （包含）

principle (原理), 189

region (区域). 378

independent (无关), 3

index (指标)

of an eigenvalue (特征值的), 131, 139, 148

ofnilpotence (幂零).37

induced matrix norm (诱导矩阵范数), 292

by absolute vector norm (由绝对范数), 310

by monotone vector norm (由单调范数). 310, 365

characterization (特征), 302, 307

inequality (不等式)

arithmetic geometric mean (算术-几何平均), 535

between matrix norms (矩阵范数间的), 314

between vector norms (向量范数间的), 279

bilinear (双线性), 473

Cauchy-Schwarz. 15, 261, 277, 535, 536

determinant (行列式), 351

Fischer, 178

Greub and Rheinboldt (Greub & Rieunboldt), 152

Grunsky. 202

Hadamard, 199, 200, 177, 183

Hadamard Fischer. 185

Holder. 276.535

Kantorovich, 444, 451, 452

matrix norm (矩阵范数), 290, 312

Minkowski, 265, 536

numerical radius (数值半径), 331

Oppenheim, 480

Ostrowski Taussky. 468, 481

positive definite function (1) 400

power for numerical radius (关于数值半径的幂). 333, 331

rank (秩), 352

Robertson, 468

square root continuity (平方根连续(函数)), 411

submultiplicative (次乘性), 290

Szasz. 479

triangle (三角), 259, 290

unitarily invariant matrix norm (两不变矩阵范数), 450

unitarily invariant norms (两不变向量范数). 147

Wieland, 112.443

inertia of a matrix (矩阵的惯性), 221

infinite series of matrices (矩阵的无穷级数). 300

inner product (内积). 140

characterization of norm derived from (由内积诱导的范数的特征), 263

definition (的定义), 260

Frobenius, 332

standard (标准), 14

usual (普通), 14

interior point (内点). 282

interlacing （交错）

eigenvalues theorem for bordered matrices (加边矩阵的特征值定理), 185

inequalities (不等式), 182, 185, 187, 189.

1. 104, 419

property for singular values (奇异值的性质), 419

theorem (定理), 182

invariant (不变)

factors (因子), 154

subspace (子空间), 51

inverse (逆), 14

diagonal dominance (对角占优(矩阵)的逆), 355

errors in (矩阵)逆的误差), 335

irreducibly diagonally dominant (不可约对角占优 (矩阵)的逆). 363

minors of (逆的子式), 21

series for (逆矩阵的级数), 301

small rank adjustment (小秩修正矩阵的逆), 18

strict diagonal dominance (严格对角占优(矩阵)的逆), 302, 349

invertible (可逆). 14

irreducible matrix (不可约矩阵), 361, 362, 493, 506-515

minimal polynomial criterion (的极小多项式准则), 515

irreducible normal form (不可约正规形式), 506

irreducible diagonally dominant (不可约对角占优). 362

isometry (等距), 68

for a vector norm (关于向量范数), 266

isomorphism (同构), 4

J

Jacobi

identity (恒等式), 21

method (法), 76

Jordan

block (块), 121

normal form (法式), 121

Jordan canonical form (Jordan 标准形), 121, 129

real (实), 152

theorem (定理), 126

K

Kakcya's theorem (Kakcya定理), 318

kernel (核), 456, 462

Kojima's bound on zeroes (零点的 Kojima 界). 319, 364

Krein Milman theorem (Krein Milman 定理), 533, 534

Kronecker product (Kronecker 乘积), 474, 475

Krylov sequence (Krylov序列), 164

L

Lagrange

equations (方程组), 227

interpolating polynomial (插值多项式), 29, 188, 405

interpolation (插值法), 29

interpolation formula (插值公式), 30

Lanczos tridiagonalization (Lanczos 三对角化, 164

Laplace

equation (方程). 239

expansion (展开式), 7

least squares (最小二乘)

approximation (逼近), 429, 431, 515

solution (解), 421

left Perron vector (左 Perron 向量), 497

Levy Dcsplanques

condition for invertibility (可逆性条件), 302, 349

theorem (定理), 302, 349

limit（极限）

of a sequence (序列的), 270

point (点). 282

line segment（线段），289

linear（线性）

dependence（相关），3

independence(无关), 3, 407

transformation(交换). 5

loop（圈），358

M

majorization (优化), 199, 425, 446

and unitarily invariant norms (与西不变范数), 447

characterizations (特征), 197

definition (定义), 192

eigenvalues by diagonal entries (方阵) 对角元组

成的向量优化（其）特征向量），193，196

product inequality (乘积不等式), 199

spectrum of a sum (矩阵) 和的谱), 194

Markov chain(Markov 钝), 497

Mason and Carmichael's bound on zacrocs (零点的

Mason和Carmichael界）.317，318，364

matrix (矩阵)

adjacency (邻接), 168, 523

almost diagonalizable (几乎可对角化), 89

approximation problems (逼近问题), 427

backward identity (后向单位), 28, 207

block diagonal (分块对角), 24

block triangular (分块角), 25. 90

bordered (加边), 185

change of basis (基变换), 32

circulant (轮换). 26

combinatorially symmetric (组合对称). 523

commuting (交换), 135

companion (友), 147, 149, 316

complex orthogonal (复正交)71, 72

complex symmetric (复对称). 201

compound (复合), 19

ronvergent (收敛), 137

correlation (相关), 400

covariance (协方差), 219, 239, 392, 424

diagonal (对角). 23

diagonalizable (可对角化), 46, 139, 145

doubly stochastic (双随机), 197, 527-529

equivalent (等价). 164

essentially nonnegative (本性非负), 506

essentially triangular (本性三角), 26

function of a (函数), 300

Gram, 407

Hankel, 27, 202, 393

Hermatian. 109, 167, 169

Hessenberg, 28

Hessian, 392, 459, 534

Hilbert, 341, 401

identity (单位), 6

indefinite (不定), 397

indicator (指标), 356

irreducible (不可约), 361, 362

Jacobian, 218

Jordan. 121, 129

negative definite (负定), 397

nilpotent (幂零), 139

nonderogatory (非减次). 135

normal (正规). 100

normal skew-symmetric (正规斜对称), 217

normal symmetric (正规对称), 207

orthogonal (1. 71, 72

orthogonally diagonalizable (正交对角化), 211

orthostochastic (正交随机), 197

permutation (置换), 25

rank one (秩1). 61

real orthogonal (实正交), 66, 72, 107

real skew-symmetric (实斜对称), 107

real symmetric (实对称), 107

reducible (可约), 360

scalar (纯量). 23

similarity (相似), 14

skew-Hermitian (斜Hermite), 100, 169

skew-orthogonal (斜正交), 72

skew-symmetric (斜对称), 109

skew-symmetric normal (斜对称正规), 217

stochastic (随机), 526-529

symmetric (对称), 167, 201

symmetric diagonalizable (对称可对角化), 211

symmetric normal (对称正规). 207

symmetric unitary (对称酉). 215

Toeplitz, 27. 136, 137, 394, 409, 456, 462

riangular (三角), 24

ridiagonal (二对角), 28, 174, 395, 409, 506

unitary (西), 66, 109

unitary characterizations (的西特征), 67

unitary symmetric (西对称), 215

Vandermonde, 29

weakly irreducible (弱不可约). 383

matrix functions, generalized (广义) 矩阵函数), 8

matrix norm (矩阵范数)

bound norm (有界范数). 294

Euclidean norm (Euclid 范数), 291

Frobenius norm (Frobenius 范数), 291

generalized （广义）：320 342

Hilbert-Schmidt norm (Hilbert-Schmidt 范数), 291

induced (诱导), 307, 365

induced by a similarity (由相似诱导的), 296

induced by vector norm (由向量范数诱导的),

1. 294

inequality for (关于不等式), 312

$l_{1}$ norm (1. 范数), 291

$\pmb{I}_{\cdot}$ norm (12 范数). 291

$n! \mod n$ （204 (2) 292

lub norm (lub 范数). 294

maximum column sum norm (极大列和范数). 294

maximum row sum norm (极大行和范数), 295

minimal (极小), 306, 307

not convex set (的非凸集), 312

operator norm (算子范数). 294

Schatten p norm (Schattenp 范数), 441

Schur norm (Schur范数), 291

self adjoint (自伴). 309

spectral norm (谱范数). 295, 441

trace norm (迹范数). 441

unitarily invariant (西不变), 292, 296, 308

max min theorem （极大极小定理）, 179, 493, 196, 504

maximal element (极大元), 384

maximum of a continuous function (连续函数的极大值). 541

McCoy's theorem (McCoy定理), 94, 97

Mercer's theorem (Mercer定理), 456

metric convex hull (度量凸包), 289

min-max theorem （极小极大定理），179，193，496

minimal polynomial (极小多项式), 142, 145

algorithm to compute (的算法), 144, 148

definition (的定义). 143

diagonalization criterion (对角化的极小多项式准则), 145

of a direct sum (直和的), 149

minimally spectrally dominant (极小谱优势). 330

Minkowski determinant inequality (Minkowski行列式不等式). 182

minor (子式), 17

principal (主), 17. 40, 398

signcd (带正负号), I7

moments of eigenvalues (特征值的矩), 43, 44

moment sequence (矩序列)

Hausdorff, 393

Toeplitz, 393

moments (矩)

algebraic (代数), 393

trigonometric (三角), 393, 455, 456

monotonicity theorem (单调性定理), 182

Montel's bound on zeros （多项式）零点的

Montel界），317，318，364

Moore Penrose generalized inverse (Moore Penrose广义逆). 121

multilinear (多重线性), 11

N

Nchari's theorem (Nchari定理), 202

nodes (结点), 168

nondefective (非亏损), 58, 103

nonderogatory (非减次), 58, 135, 147

nonnegative (非负), 259, 260, 290

nonnegative matrix (非负矩阵), 359

applications (的应用), 487-490

definition (的定义), 490

doubly stochastic (双随机), 527-529

eigenvalues (特征值), 189, 503-505, 507-515

cigenvectors （特征向量）.489，503-505，507-515

general limit theorem (一般极限定理), 524

irreducible (不可约), 507-515

limit of powers (幂的极限), 489

limit theorems (极限定理), 500, 516, 524, 525

Perron root (Perron 根), 505, 508

Perron vector (Perron 向量), 505. 508

Perron-Frobenius theorems (Perron-Frobenius 定理), 508-511

primitive matrices (素矩阵), 515-524

spectral radius (谱半径), 489, 491-495, 503-505, 507-515

stochastic (随机), 526-529

non-singularity. characterizations of (非奇异性的特征), 14

nontrivial cycle (非平凡回路), 383

norm （范数）

and inversion (与可逆性), 301

characterization of derived (诱导范数的特征), 263

compatible (相容), 294, 324

consistent (协和). 324

dual (对偶), 275, 410

dual pair (对偶对), 278

minunally spectrally dominant (极小谱优势), 330

pre norm (准范数), 272

spectrally dominant (谱优势), 324

subordinate (从属), 324

normal matrix (正规矩阵)

characterizations (特征), 101, 108-112

definition (定义), 100

perfectly conditioned eigenvalues （优态特征值），367

real (实). 104ff

null space (零空间), 5, 262

numerical radius (数值半径), 321, 331, 332, 333, 334

numerical range (数值范围), 321

0

open set (开集), 282, 541

operator norm (算子范数). 294

orthogonal (正交)

complement (分量). 16

vectors (向量), 15

orthogonality (15)

orthogonally （正交）

diagonalizable (可对角化). 101

equivalent (等价), 73

orthonormal (标准正交), 15

basis (基). 16

Ostrowski

condition for invertibility (可逆性条件), 381

region (区域), 378, 379

theorem (定理), 224, 378

ovalsofCassini (Cassini椭圆形).380

P

partitioned matrix (分块矩阵)

definition (的定义), 17

inverse (18, 472

Schur complement, 472

Pearcy's theorem (Pearcy定理), 76

perfectly conditioncd (优态), 336

permanent (积利式). 8

permutation invariant (置换不变(函数))，368，438

Perron

root（根），497，505，508

theorem (定理), 500

vector (向量), 497, 505, 508

Perron-Frobenius theorems (Perron-Frobenius定理).508-511

perturbation (扰动)

cigenvalues (特征值), 198, 343, 364

linear equations (线性方程组). 335

theorems (定理), 364

plane rotations (平面旋转), 74

Poincaré separation theorem (Poincaré分离定理), 190, 141

polar (极)

coordinates in complex plane (复平面的极坐标), 532

decomposition (极分解)

polar form (极形式), 156, 411, 412ff

examples and applications (例及应用), 427ff

polynomial (多项式)

bounds for zeroes of (零点的界), 316-319

continuous dependence of zeros on coefficients

（的零点对（其）系数的连续依赖），539

for inverse (逆矩阵的). 88

in a matrix (矩阵的), 36, 135, 142

monic（首·），142

similarityinvariants (相似不变量),154

zeros of a (的零点), 537

zeros of a real (实的零点), 538

poorlyconditioned（坏条件），336

positive (iF(定)), 259, 260, 290

cone (锥), 398

definite function (函数), 400, 401, 163

definite kernel (核), 402, 462

positive definite matrix (正定矩阵), 250

applications (的应用), 391-396, 459

characteristic polynomial (的特征多项式), 403

characterizations (402)

concavity of logarithm of the determinant (行列式的对数凹性), 466

concavity of trace of the inverse (的逆的迹的四性), 468

definition (定义), 396

determinant criterion (行列式判别准则), 404

determinantal inequalities (行列式不等式), 476 486

cigenvalues (的特征值), 402

kth root (k次根), 405

ordcring（次序关系）, 469ff

positivematrix (正矩阵), 359

definition (的定义), 490

eigenvalues (特征值), 495-503

cigenvectors (特征向量), 495-503

left Perron vector (左 Perron向量), 497

Perron root (Perron 根), 497

Perron vector (Perron 向量), 497

Perron's theorem (Perron定理), 500

spectral radius (谱半径), 495-503

positivc semidefinite matrix (半正定矩阵), 182

definition (定义), 396

kth root (k次根), 405

ordering (次序关系). 469ff

rank $k$ （秩 $\pmb{k}$ ），457

positive semidefinite ordering (半正定次序关系), 469

power method (幂法), 62, 523

preorder (预序), 384

primitive matrix (素(或本原)矩阵), 515-524

characterizations (的特征), 516, 517

definition (的定义), 516

directed graph (的有向图), 517

eigenvalues (的特征值), 516

Holloday-Varga theorem (的 Holloday-Varga 定理), 520

index of primitivity (的本原指标), 519

Wielandt's theorem (Wielandt定理), 520

procrustean transformation (强行变换), 431

property (性质)

L. 97. 100

P. 100

SC，355，358，359，362

pure imaginary complex number (纯虚复数), 532

Q

QR

algorithm (算法), 114

factorization (分解), 112, 164, 406

quasi-linearization (拟线性化), 191, 453, 455, 486

R

range (值域), 5

rank (秩). 12

equalities (秩的等式), 13

incqualities (秩的不等式), 13, 175, 352, 458

rank one (秩1)

approximation (逼近), 127

limit (极限), 499

matrix(矩阵), 61

rational（有理）

canonical form(标准形), 156

form (形), 154

Rayleigh Ritz

ratio (比). 176

theorem (定理), 176, 422

real (实)

axis (实轴), 532

Jordan canonical form (实 Jordan 标准形), 152

part of a complex number (复数的实部), 531

rectangular coordinates in complex plane (复平面中的直角坐标), 532

residual vector (剩余向量), 338, 373, 374

reverse order law (倒序律). 6

Riemann sum (Riemann 和), 462

right （右）

eigenvector (右特征向量), 59

half-plane (右半平面), 532

Romanovsky's theorem (Romanovsky定理), 517

rotation problem (旋转问题). 431, 435

round-off errors (舍入误差), 335

row rank (行秩), 12

row-reduced echelon form (RREF) (行简化梯形阵), 10

s

scalar product (纯量积), 14

Schur

complement (补), 21, 472

majorization theorem (优化定理), 193

norm (范数). 291

product （乘积）

product theorem (乘积定理), 455, 458

unitary triangularization theorem (两三角化定理), 79, 83

self-adjoint vector norm on matrices (关于矩阵的自伴向量范数), 450

sensitivity (灵敏度)

of eigenvalues (特征值的), 372

of eigenvectors (特征向量的), 373

of solutions of linear equations (线性方程组解的), 339

separating hyperplane theorem (分离超平面定理). 534

shift operator (移位算子), 38

signature of a matrix (矩阵的符号差), 221

signum（sgn） （正负号），8

similarity (相似(性)). 44

inverse to adjoint (逆相似于伴随), 70

matrix and its transpose (矩阵与其转置). 134

to a real matrix (相似于实矩阵), 172

to a real matrix, $A\overline{A}$ (AA相似于实矩阵), 253

to adjoint (相似于伴随), 172

simultaneous (同时)

congruence, canonical pairs (‘相合下的标准形偶), 236

condiagonalization (合对角化), 252

simultaneous diagonalization (同时对角化), 49, 52

by "congruence (用"相合), 240, 464ff

by congruence (用相合), 228, 241, 250

by unitary congruence (用酉相合), 228, 235

characterization of (的特征), 228

singular (奇异), 14

singular value decomposition (奇异值分解), 157, 205, 325, 411, 414ff, 421

examples and applications (的例及应用), 427ff

simultaneous (同时), 426

singular values (奇异值), 205. 415

largest (最大), 421

of a product (乘积的), 423

of a sum (和的). 423

perfectly conditioned (优态), 418

variational characterization (420)

skew Hermitian (斜Hermite)

matrix (矩阵), 100, 169

part (部分), 109, 170, 399

solution equivalent systems (解等价方程组), 11

span (张成)2，3

Specht's theorem (Specht定理), 76

spectral characteristic (谱示性数), 330

spectral radius (谱半径). 35, 198, 296, 313. 348, 489

as a limit of matrix norms (作为矩阵范数极限的), 297

as a limit using norms or pre norms, （作为用范数或准范数表示的极限的）, 299, 322

spectral theorem (谱定理)

Hermitian matrices (Hermite 矩阵的), 104, 171

normal matrices (正规矩阵的), 101, 425

spectrally dominant (谱优势), 329

spectrum (谱), 35

of a sum (矩阵) 和的), 92

of a sum by majorization (用优化表示和的谱), 194

square root of a matrix (矩阵的平方根), 54, 405

continuity of (的连续性), 411

standard (标准)

basis (基), 4

inner product (内积), 14

strictly diagonally dominant (严格对角占优(矩阵), 302, 349

strongly connected directed graph (强连通有向图), 358, 362, 383

submatrix (子矩阵). 4

eigenvalues (的特征值), 189

principal (主), 17, 397

submultiplicative (次乘性), 290

subspace (子空间), 2

invariant (不变), 51

Sylvester's law of inertia (Sylvester 惯性定律),

1. 238

analoguc for symmetric matrices (对称矩阵的模拟), 225

homotopy proof (同伦证明). 242

symmetric (对称)

gauge function (度规函数), 438, 445

symmetric matrix (对称矩阵). 167, 397

diagonalizable (对角化), 211

every matrix similar to a (每个矩阵相似于一个), 209

product of two (两个对称矩阵的乘积), 210

real（实），169，218

Szasz's inequality (Szasz 不等式), 479

T

Takagi's factorization (Takagi分解), 204, 423, 466

asconsimilarityanalogueofspectraltheorem（类似于谱定理的合相似下的），250

Hua'sproof (Hua证明),217

Siegel'sproof （Siegcl证明），216

Tausky's theorem (Taussky定理). 363

topological notions (拓扑概念), 282, 288

trace (迹). 40, 175, 398

norm (迹范数), 441, 445

zero (迹为零), 77

transpose (转置(矩阵)), 6

transposition (对换(矩阵)). 25

triangularization (三角化)

byconsimilarity (用合相似),244,245

by unitary congruence (用酉相合), 203

by unitaryconsimilarity (用西合相似),244,245

by unitary similarity (用酉相似), 79

orthogonal (正交), 84

simultaneous (同时), 81, 84, 94

tripotent (三次幂等), 148

trivial cycle (平凡回路), 358

truncation errors (截断误差), 335

U

unit （单位）

disc (圆盘), 532

sphere (球面). 273

unit ball (单位球), 273, 281

compact (紧), 283, 284

convex (p), 284

equilibrated (均衡), 284

properties of (的性质), 283

unitarily diagonalizable (西可对角化), 10!

unitarilyequivalent （两等价）

definition (定义), 72

equal diagonal entries (对角元相等的矩阵). 77

Specht's theorem (酉等价的 Specht 定理), 76

to upper triangular matrix (于上三角矩阵), 79

unitarily invariant matrix norms (酉不变矩阵范数), 296, 308

set is convex (之集是凸集), 450

unitarily invariant vector norm. on matrices (关于矩阵的酉不变向量范数), 437, 441, 445

Von Neumann's characterization (the Von Neumann特征), 438

when a matrix norm (何时是矩阵范数), 450

unitary group (两群)

unitary matrix (酉矩阵), 66-72, 157

characterizations (67

definition (的定义). 66

selection principle (的选择原理). 69

upper half-plane (上半平面), 532

V

variational characterization of eigenvalues (特征值的变分特征), 176

vector (向量)

normalized (正规化), 15

unit (单位), 15

vector norm (向量范数)

absolute (绝对), 285, 310, 365, 438

algebraic properties (代数性质), 268

analytic properties (分析性质), 269

Cauchy sequence with respect to a (关于向量范数的 Cauchy序列), 274

characterization via unit ball (的单位球的特征). 284

compatible (相容). 324. 327

consistent (协和), 321

convergence with respect to a (关于向量范数收敛), 269

definition (的定义), 259

derived from inner product (由内积诱导的), 262

dual (对偶), 275

duality theorem (对偶性定理), 287

equivalent (的等价). 273. 279

Euclid (Euclid), 264

geometric properties (几何性质), 281

$\pmb{I}_{1}$ norm ( $\pmb{I}_{1}$ 范数），263

$l_{\cdot}$ norm (t范数).264

$I_{p}$ norm ( $\ell_p$ 范数）.265

l norm (1. 范数), 265, 322

$L_{1}$ norm (L范数), 266

$L_{2}$ norm (范数). 266

$L_{p}$ norm ( $L_{p}$ 范数), 267

l. norm (L. 范数), 267

Manhattan norm (Manhattan 范数), 265

max norm (极大范数), 265

monotone (单调), 285, 310, 365, 449

on matrices (关于矩阵的), 320-342

polyhedral (多面), 282

pre norm (准范数), 272, 322

spectrally dominant (谱优势), 329

subordinate (从属), 324

sum norm (相), 265

uniformly continuous (一致连续(性)), 271

unit ball (单位球), 273, 281

unit sphere (单位球面), 273

unitarily invariant (西不变), 265, 267

unarily invariant on matrices (矩阵的酉不变), 437

weakly monotone (弱单调), 285

vector seminorm (向量半范数), 250

vector space (向量空间). 2

complete (完备), 271

Von Neumann

inner product theorem (内积定理), 263

unitarily invariant norm theorem (两不变范数定理), 438

W

weak minimum principle (弱极小原理), 460

weakly connected directed graph (弱连通有向图), 383

Weierstrass's theorem (Weierstrass 定理). 541

weighted arithmetic-geometric mean inequality (加权算术几何平均值不等式), 535

well conditioned (良态的), 336

Weyl's theorem (Weyl定理), 181, 184, 367, 419, 423

Wielandt

example （例）.522

theorem (定理), 520

Witt cancellation theorem (Witt 消去定理). 78.141