Basic matrix algebra with algorithms and applications:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton, Fla. [u.a.]
Chapman & Hall/CRC
2003
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| Schriftenreihe: | Chapman & Hall/CRC mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 242 S. graph. Darst. |
| ISBN: | 1584883332 |
Internformat
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Datensatz im Suchindex
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| adam_text | BASIC MATRIX ALGEBRA WITH ALGORITHMS AND APPLICATIONS ROBERT A. LIEBLER
CHAPMAN & HALL/CRC A CRC PRESS COMPANY BOCA RATON LONDON NEW YORK
WASHINGTON, D.C. CONTENTS PREFACE XI EXAMPLES XIII MAJOR RESULTS/PROOFS
XV 1 SYSTEMS OF LINEAR EQUATIONS AND THEIR SOLUTION 1 1.1 RECOGNIZING
LINEAR SYSTEMS AND SOLUTIONS 2 GIVEN A SYSTEM OF EQUATIONS AND A
COLLECTION OF VARIABLES, DETERMINE WHETHER THE SYSTEM IS LINEAR IN THE
GIVEN VARIABLES 2 GIVEN A SYSTEM OF LINEAR EQUATIONS IN N VARIABLES,
DETERMINE WHETHER A GIVEN RA-TUPLE IS A SOLUTION 3 1.2 MATRICES,
EQUIVALENCE AND ROW OPERATIONS 7 GIVEN A SYSTEM OF LINEAR EQUATIONS WITH
NUMERICAL COEFFICIENTS, WRITE THE ASSOCIATED AUGMENTED MATRIX AND VICE
VERSA 7 GIVEN A MATRIX AND SEQUENCE OF ELEMENTARY ROW OPERATIONS, APPLY
THE ROW OPERATIONS TO OBTAIN AN EQUIVALENT MATRIX 10 1.3 ECHELON FORMS
AND GAUSSIAN ELIMINATION 17 GIVEN A MATRIX, DECIDE IF IT IS IN REDUCED
ROW ECHELON FORM 17 GIVEN A MATRIX, APPLY ELEMENTARY ROW OPERATIONS TO
OBTAIN AN EQUIVALENT MATRIX IN REDUCED ROW ECHELON FORM 18 SOLVE SEVERAL
LINEAR SYSTEMS WITH THE SAME COEFFICIENT MATRIX AT ONCE. ... 22 1.4 FREE
VARIABLES AND GENERAL SOLUTIONS 25 GIVEN AN RREF AUGMENTED MATRIX, WRITE
THE GENERAL SOLUTION OF THE AS- SOCIATED LINEAR SYSTEM IN TERMS OF THE
VARIABLES FROM NONPIVOTAL COLUMNS OF THE COEFFICIENT MATRIX, OR STATE
THAT THERE IS NO SOLUTION. 25 GIVEN A SYSTEM OF LINEAR EQUATIONS, WRITE
AN ASSOCIATED AUGMENTED MATRIX, APPLY ELEMENTARY ROW OPERATIONS TO
OBTAIN AN EQUIVALENT RREF MA- TRIX AND WRITE THE GENERAL SOLUTION* 27
1.5 THE VECTOR FORM OF THE GENERAL SOLUTION 30 GIVEN A MATRIX A AND A
VECTOR X COMPUTE AX IF IT EXISTS. TRANSLATE BETWEEN A LINEAR SYSTEM AND
THE ASSOCIATED MATRIX EQUATION 30 COMPUTE EXPRESSIONS AND VERIFY
IDENTITIES USING MATRIX-VECTOR MULTIPLICA- TION AND LINEAR COMBINATIONS
31 GIVEN AX. = B, WITH A IN AN ECHELON FORM, WRITE THE GENERAL SOLUTION
AS A LINEAR COMBINATION OF BASIC SOLUTIONS AND THE DISTINGUISHED SOLU-
TION 34 VN VIII CONTENTS. 1.6 GEOMETRIC VECTORS AND LINEAR FUNCTIONS 38
INTERPRET VECTOR ARITHMETIC GEOMETRICALLY. DISTANCES AND ANGLES IN R 2 ,
R 3 . . 38 TRANSFORM PLOTS OF LINEAR FUNCTIONS AND LINEAR SYSTEMS
GEOMETRICALLY. ... 41 1.7 POLYNOMIAL INTERPOLATION 49 DETERMINE THE
GENERAL FORM OF A POLYNOMIAL FUNCTION HAVING DEGREE AT MOST N AND HAVING
M SPECIFIED FUNCTIONAL VALUES, W# 3 49 2 MATRIX NUMBER SYSTEMS 53 2.1
COMPLEX NUMBERS 54 COMPUTE SUMS, DIFFERENCES, PRODUCTS AND QUOTIENTS IN
C. COMPUTE THE ZEROS OF QUADRATIC POLYNOMIALS 54 INTERPRET COMPLEX
ARITHMETIC IN THE GEOMETRIC REPRESENTATION USING BOTH THE ( RECTANGULAR
AND POLAR FORMS 56 2.2 MATRIX MULTIPLICATION 62 GIVEN MATRICES, COMPUTE
SUMS, SCALAR MULTIPLES AND PRODUCTS AS INDICATED, OR DECIDE THE
INDICATED EXPRESSION DOES NOT EXIST 62 GIVEN AN ADJACENCY MATRIX, SKETCH
THE ASSOCIATED LABELED DIGRAPH AND VICE VERSA. USE MATRIX MULTIPLICATION
TO COMPUTE THE NUMBER OF WALKS OF SPECIFIED TYPE 63 SIMPLIFY AND
EVALUATE MATRIX EXPRESSIONS USING PROPERTIES OF MATRIX ARITH- METIC 64
VERIFY COMPLEX IDENTITIES IN THE MATRIX REPRESENTATION OF C 66 2.3
AUXILIARY MATRICES AND MATRIX INVERSES 69 GIVEN A POLYNOMIAL F(X) A
SQUARE MATRIX A AND A VECTOR V, COMPUTE /( A) V WITH AN AUXILIARY MATRIX
69 GIVEN AN RA-BY-N MATRIX A, COMPUTE A~ L OR DECIDE THAT A ISN T
INVERTIBLE. . 72 2.4 SYMMETRIC PROJECTORS, RESOLVING VECTORS 77 GIVEN AN
N-BY-R MATRIX A OF RANK R, COMPUTE PA AND THE COMPLEMENTARY PROJECTOR P
A . EXPLAIN THEIR GEOMETRIC MEANING 77 GIVEN VECTORS A, W G R 3 ,
RESOLVE W WITH RESPECT TO A 79 2.5 LEAST SQUARES APPROXIMATION 82 GIVEN
A SET OF UP TO FIVE DATA POINTS AND M 4, FIND THE (M * L)-TH DEGREE
LEAST SQUARES APPROXIMATION F(X) TO THE DATA 82 2.6 CHANGING PLANE
COORDINATES 87 GIVEN TWO PLANE COORDINATE SYSTEMS SPECIFIED BY THEIR
ORIGINS AND BASIC UNIT VECTORS, COMPUTE THE COORDINATES OF A POINT
SPECIFIED IN ONE COORDI- NATE SYSTEM IN TERMS OF THE OTHER. 87 2.7 THE
FAST FOURIER TRANSFORM AND THE EUCLIDEAN ALGORITHM 95 GIVEN A POLYNOMIAL
G(X) AND AN INTEGER N DEG(G), WRITE THE MATRIX R G OF THE REMAINDER
WHEN DIVIDED BY G(X). 95 GIVEN A ROW PARTITIONED VANDERMONDE MATRIX,
FACTOR IT AS A BLOCK DIAGONAL MATRIX OF SMALLER VANDERMONDE MATRICES AND
OF STACKED REMAINDER MATRICES. R . 98 GIVEN TWO POLYNOMIALS, F(X),G(X)
COMPUTE POLYNOMIALS A(X),B(X) SUCH THAT A(X)F(X) + B(X)G(X) EQUALS THEIR
GREATEST COMMON DIVISOR. . 102 CONTENTS. IX DIAGONALIZABLE MATRICES 109
3.1 EIGENVECTORS AND EIGENVALUES 110 GIVEN A SQUARE MATRIX A AND VECTOR
V, DECIDE IF V IS AN EIGENVECTOR OF A. . 110 GIVEN A SQUARE MATRIX A AND
A NUMBER A, DECIDE IF A IS AN EIGENVALUE OF A, AND IF SO, FIND THE
ASSOCIATED BASIC EIGENVECTORS 112 3.2 THE MINIMAL POLYNOMIAL ALGORITHM
117 GIVEN A SQUARE MATRIX A, A VECTOR V AND A POLYNOMIAL F(X), DECIDE IF
F(X) ANNIHILATES THE PAIR A, V. IF NOT, DECIDE THE A ANNIHILATOR OF V. .
. 117 COMPUTE THE MINIMAL POLYNOMIAL OF A SQUARE MATRIX 121 3.3 LINEAR
RECURRENCE RELATIONS 128 GIVEN A DIGRAPH T AND A SPECIFIED WALK TYPE,
LET W N BE THE NUMBER OF WALKS OF LENGTH N IN V AND OF THE SPECIFIED
TYPE. COMPUTE THE INITIAL VALUES AND A RECURRENCE RELATION SATISFIED BY
THE SEQUENCE W N . . . 128 LET B N BE THE NUMBER OF BINARY SEQUENCES OF
LENGTH N HAVING NO SUBLISTS IN A GIVEN LIST L AND STARTING AND ENDING IN
SPECIFIED WAYS. DETERMINE THE INITIAL VALUES OF B N AND A RECURRENCE
RELATION SATISFIED BY {6 N } USING A TRANSFER MATRIX 132 3.4 PROPERTIES
OF THE MINIMAL POLYNOMIAL 136 GIVEN AN N-BY-N MATRIX (N 3), WRITE AP =
PD, WHERE D IS DIAGONAL AND P HAS COLUMNS THE STANDARD LIST OF BASIC
EIGENVECTORS OF A. . . 136 GIVEN A DIAGONALIZABLE MATRIX A AND ITS
MINIMAL POLYNOMIAL, WRITE A AS A SUM OF ITS COMPONENTS 141 3.5 THE
SEQUENCE {A K } 147 GIVEN AN N-BY-N MATRIX A, N 3, DECIDE IF IT IS
DIAGONALIZABLE AND IF SO, FIND ITS SPECTRAL RADIUS AND A FORMULA FOR AN
ARBITRARY POWER OF A. . 147 GIVEN AN N-BY-N MATRIX A, N 3, DESCRIBE
THE EVOLUTION OF {A K } 148 GIVEN A 3-BY-3 NUMERICAL MATRIX WITH THREE
DISTINCT REAL EIGENVALUES, COM- PUTE THE DOMINANT EIGENVALUE BY
ITERATION AND GIVE THE DEFLATED MA- TRIX 152 3.6 DISCRETE DYNAMICAL
SYSTEMS 158 GIVEN A THREE-STATE LINEAR DISCRETE DYNAMICAL SYSTEM, WRITE
ITS TRANSITION MA- TRIX A. COMPUTE THE SPECTRAL RADIUS OF A AND
DETERMINE THE SYS- TEM S LONG-TERM BEHAVIOR. IDENTIFY ANY STABLE STEADY
STATES 158 3.7 MATRIX COMPRESSION WITH COMPONENTS 167 GIVEN R 2 AND A
4 BY 4 MATRIX WITH DISTINCT EIGENVALUES, COMPUTE ITS COMPONENTS AND
WRITE AN OPTIMAL RANK R APPROXIMATION TO A. . . 167 GIVEN DATA INVOLVING
THREE ATTRIBUTES, STANDARDIZE IT AND COMPUTE THE ASSO- CIATED
CORRELATION MATRIX. DETERMINE THE FIRST PRINCIPAL COMPONENT AND
DETERMINE WHAT PERCENTAGE OF THE DATA VARIANCE IT EXPLAINS. . . 169 X
CONTENTS. 4 DETERMINANTS 177 4.1 AREA AND COMPOSITION OF LINEAR
FUNCTIONS 178 GIVEN SQUARE MATRIX A, FACTOR IT AS A PRODUCT OF
ELEMENTARY MATRICES AND RREF(A). USE A SEQUENCE OF TRANSFORM PLOTS TO
REALIZE THE ASSOCI- ATED LINEAR FUNCTION /A AS A COMPOSITION OF
ELEMENTARY LINEAR FUNC- TIONS 178 GIVEN A TRANSFORM PLOT /A : R 2 * R
2 THAT TRACKS A FIGURE, USE AREA AND ORIENTATION TO COMPUTE DET(A) 180
4.2 COMPUTING DETERMINANTS 185 COMPUTE THE DETERMINANT OF AN N-BY-N(N
4) MATRIX USING THE COFACTOR DEFINITION 185 COMPUTE THE DETERMINANT OF
AN N-BY-N(N 4) MATRIX USING GAUSSIAN ELIM- INATION AND THE BLOCK
TRIANGULAR FORMULA 189 4.3 FUNDAMENTAL PROPERTIES OF DETERMINANTS 196
COMPUTE THE DETERMINANT OF AN N-BY-N (N 5) MATRIX USING ARBITRARY CO-
FACTOR EXPANSIONS, MATRIX REDUCTION OR BLOCK MATRICES AS REQUIRED. . 196
USE PROPERTIES OF DETERMINANTS TO VERIFY FORMULAS WITHOUT COMPUTATION. .
. 199 GIVEN A 2-BY-2 INTEGER MATRIX, COMPUTE ITS SMITH NORMAL FORM AND
INTERPRET THE ASSOCIATED LATTICE GRAPHICALLY 202 4.4 FURTHER
APPLICATIONS 207 FIND THE CHARACTERISTIC POLYNOMIAL/EQUATION OF AN
N-BY-N MATRIX (N 4). . 207 USE CRAMER S RULE TO SOLVE THE LINEAR
SYSTEM AX = B WHEN APPROPRIATE. . . 209 A PEDAGOGICAL POSTSCRIPT 213
APPENDIX: THE ABSTRACT SETTING 217 SELECTED PRACTICE PROBLEM ANSWERS 231
INDEX 239
|
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| author | Liebler, Robert A. |
| author_facet | Liebler, Robert A. |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.9434 |
| dewey-search | 512.9434 |
| dewey-sort | 3512.9434 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| spelling | Liebler, Robert A. Verfasser aut Basic matrix algebra with algorithms and applications Robert A. Liebler Boca Raton, Fla. [u.a.] Chapman & Hall/CRC 2003 XV, 242 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC mathematics Matrix Mathematik (DE-588)4037968-1 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 s DE-604 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017434686&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Liebler, Robert A. Basic matrix algebra with algorithms and applications Matrix Mathematik (DE-588)4037968-1 gnd |
| subject_GND | (DE-588)4037968-1 |
| title | Basic matrix algebra with algorithms and applications |
| title_auth | Basic matrix algebra with algorithms and applications |
| title_exact_search | Basic matrix algebra with algorithms and applications |
| title_full | Basic matrix algebra with algorithms and applications Robert A. Liebler |
| title_fullStr | Basic matrix algebra with algorithms and applications Robert A. Liebler |
| title_full_unstemmed | Basic matrix algebra with algorithms and applications Robert A. Liebler |
| title_short | Basic matrix algebra with algorithms and applications |
| title_sort | basic matrix algebra with algorithms and applications |
| topic | Matrix Mathematik (DE-588)4037968-1 gnd |
| topic_facet | Matrix Mathematik |
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