Matrica (matematika)
U matematici, matrica je pravougaona tabela brojeva, ili opštije, tabela koja se sastoji od apstraktnih objekata koji se mogu sabirati i množiti.
Matrice se koriste da opišu linearne jednačine, da se prate koeficijenti linearnih transformacija, kao i za čuvanje podataka koji zavise od dva parametra. Matrice se mogu sabirati, množiti, i razlagati na razne načine, što ih čini ključnim konceptom u linearnoj algebri i teoriji matrica.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Matrix-sr.svg/300px-Matrix-sr.svg.png)
Definicije i notacije[uredi | uredi izvor]
Horizontalne linije u matrici se nazivaju vrstama, a vertikalne kolonama matrice.[1]
Preslikavanje , takvo da je polje i nazivamo matricom tipa nad poljem F.
Matrica sa m vrsta i n kolona se naziva m-sa-n matricom (kaže se i zapisuje da je formata m×n) a m i n su dimenzije matrice.
Član matrice A, koji se nalazi u i-toj vrsti i u j-toj koloni se naziva (i,j)-ti član matrice A. Ovo se zapisuje kao Ai,j ili A[i,j]. Uvek se prvo naznačuje vrsta, pa kolona.
Često se piše kako bi se definisala m × n matrica A čiji se svaki član, A[i,j] naziva ai,j za sve 1 ≤ i ≤ m i 1 ≤ j ≤ n. Međutim, konvencija da i i j počinju od 1 nije univerzalna: neki programski jezici započinju od nule, u kom slučaju imamo 0 ≤ i ≤ m − 1 i 0 ≤ j ≤ n − 1.
Matricu čija je jedna od dimenzija jednaka jedinici često nazivamo vektorom, i interpretiramo je kao element realnog koordinatnog prostora. 1 × n matrica (jedna vrsta i n kolona) se naziva vektor vrsta, a m × 1 matrica (jedna kolona i m vrsta) se naziva vektor kolona.
Primer[uredi | uredi izvor]
Matrica
je 4×3 matrica. Element A[2,3] ili a2,3 je 7.
Matrica
je 1×9 matrica, ili vektor vrsta sa 9 elemenata.
Sabiranje i množenje matrica[uredi | uredi izvor]
Neka su date matrice i .
Sabiranje[uredi | uredi izvor]
Zbir matrica A i V, u oznaci A+V je matrica za koju važi za svako .
Množenje skalarom[uredi | uredi izvor]
Ako uzmemo matricu A i broj c, skalarni proizvod cA se računa množenjem skalarom c svakog elementa A (t. j. (cA)[i, j] = cA[i, j] ). Na primer:
Operacije sabiranja i množenja skalarom pretvaraju skup M(m, n, R) svih m-sa-n matrica sa realnim članovima u realni vektorski prostor dimenzije mn.
Međusobno množenje matrica[uredi | uredi izvor]
Množenje dve matrice je dobro definisano samo ako je broj kolona leve matrice jednak broju vrsta desne matrice. Ako je A matrica dimenzija m-sa-n, a B je matrica dimenzija n-sa-p, tada je njihov proizvod AB matrica dimenzija m-sa-p (m vrsta, p kolona) dat formulom:
za svaki par i i j.
Na primer:
Množenje matrica ima sledeća svojstva:
- (AB)C = A(BC) za sve k-sa-m matrice A, m-sa-n matrice B i n-sa-p matrice C (asocijativnost).
- (A + B)C = AC + BC za sve m-sa-n matrice A i B i n-sa-k matrice C (desna distributivnost).
- C(A + B) = CA + CB za sve m-sa-n matrice A i B i k-sa-m matrice C (leva distributivnost).
Valja znati da komutativnost ne važi u opštem slučaju; ako su date matrice A i B, čak i ako su oba proizvoda definisana, u opštem slučaju je AB ≠ BA.
Posebno, skup M(n, R) svih kvadratnih matrica reda n jeste realna asocijativna algebra sa jedinicom, koja je nekomutativna za n ≥ 2.
Linearne transformacije, rang, transponovana matrica[uredi | uredi izvor]
Matrice mogu na zgodan način da predstave linearne transformacije jer množenje matrica odgovara slaganju preslikavanja, kao što će dalje biti opisano. Upravo ovo svojstvo matrice čini moćnom strukturom podataka u višim programskim jezicima.
Ovde i u nastavku, posmatramo Rn kao skup kolona ili n-sa-1 matrica. Za svako linearno preslikavanje f : Rn → Rm postoji jedinstvena m-sa-n matrica A, takva da f(x) = Ax za svako x u Rn. Kažemo da matrica A predstavlja linearno preslikavanje f. Ako k-sa-m matrica B predstavlja drugo linearno preslikavanje g : Rm → Rk, tada je njihova kompozicija g o f takođe linearno preslikavanje Rm → Rn, i predstavljeno je upravo matricom BA. Ovo sledi iz gore pomenute asocijativnosti množenja matrica.
Opštije, linearno preslikavanje iz n-dimenzionog vektorskog prostora u m-dimenzioni vektorski prostor je predstavljeno m-sa-n matricom, ako su izabrane baze za svaki.
Rang matrice A je dimenzija slike linearnog preslikavanja predstavljenog sa A; ona je ista kao dimenzija prostora generisanog vrstama A, i takođe je iste dimenzije kao prostor generisan kolonama A.
Transponovana matrica, matrice m-sa-n, A je n-sa-m matrica Atr (nekad se zapisuje i kao AT ili tA), koja nastaje pretvaranjem vrsta u kolone, i kolona u vrste, to jest Atr[i, j] = A[j, i] za svako i i j. Ako A predstavlja linearno preslikavanje u odnosu na dve baze, tada matrica Atr predstavlja linearno preslikavanje u odnosu na dualne baze (vidi dualni prostor).
Važi (A + B)tr = Atr + Btr i (AB)tr = Btr Atr.
Vidi još[uredi | uredi izvor]
Osobine matrica[uredi | uredi izvor]
Posebne matrice[uredi | uredi izvor]
Reference[uredi | uredi izvor]
Literatura[uredi | uredi izvor]
- Ayres, Frank, Schaum's Outline of Modern Abstract Algebra, McGraw-Hill; 1st edition (June 1, 1965). ISBN 0-07-002655-6.
- Anton, Howard (1987), Elementary Linear Algebra (5th izd.), New York: Wiley, ISBN 0-471-84819-0
- Arnold, Vladimir I.; Cooke, Roger (1992), Ordinary differential equations, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-3-540-54813-3
- Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1
- Association for Computing Machinery (1979), Computer Graphics, Tata McGraw–Hill, ISBN 978-0-07-059376-3
- Baker, Andrew J. (2003), Matrix Groups: An Introduction to Lie Group Theory
, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-1-85233-470-3
- Bau III, David; Trefethen, Lloyd N. (1997), Numerical linear algebra, Philadelphia, PA: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-361-9
- Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields
, Boston: Houghton Mifflin Co., ISBN 0-395-14017-X
- Bretscher, Otto (2005), Linear Algebra with Applications (3rd izd.), Prentice Hall
- Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490
- Bronson, Richard (1989), Schaum's outline of theory and problems of matrix operations, New York: McGraw–Hill, ISBN 978-0-07-007978-6
- Brown, William C. (1991), Matrices and vector spaces
, New York, NY: Marcel Dekker, ISBN 978-0-8247-8419-5
- Coburn, Nathaniel (1955), Vector and tensor analysis, New York, NY: Macmillan, OCLC 1029828
- Conrey, J. Brian (2007), Ranks of elliptic curves and random matrix theory, Cambridge University Press, ISBN 978-0-521-69964-8
- Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd izd.), Reading: Addison-Wesley, ISBN 0-201-01984-1
- Fudenberg, Drew; Tirole, Jean (1983), Game Theory, MIT Press
- Gilbarg, David; Trudinger, Neil S. (2001), Elliptic partial differential equations of second order (2nd izd.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-3-540-41160-4
- Godsil, Chris; Royle, Gordon (2004), Algebraic Graph Theory, Graduate Texts in Mathematics, 207, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-95220-8
- Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd izd.), Johns Hopkins, ISBN 978-0-8018-5414-9
- Greub, Werner Hildbert (1975), Linear algebra, Graduate Texts in Mathematics, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-90110-7
- Halmos, Paul Richard (1982), A Hilbert space problem book, Graduate Texts in Mathematics, 19 (2nd izd.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-90685-0, MR 675952
- Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6
- Householder, Alston S. (1975), The theory of matrices in numerical analysis, New York, NY: Dover Publications, MR 0378371
- Kreyszig, Erwin (1972), Advanced Engineering Mathematics (3rd izd.), New York: Wiley, ISBN 0-471-50728-8.
- Krzanowski, Wojtek J. (1988), Principles of multivariate analysis, Oxford Statistical Science Series, 3, The Clarendon Press Oxford University Press, ISBN 978-0-19-852211-9, MR 969370
- Itô, Kiyosi, ur. (1987), Encyclopedic dictionary of mathematics. Vol. I-IV (2nd izd.), MIT Press, ISBN 978-0-262-09026-1, MR 901762
- Lang, Serge (1969), Analysis II, Addison-Wesley
- Lang, Serge (1987a), Calculus of several variables (3rd izd.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-96405-8
- Lang, Serge (1987b), Linear algebra, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-96412-6
- Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third izd.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
- Latouche, Guy; Ramaswami, Vaidyanathan (1999), Introduction to matrix analytic methods in stochastic modeling (1st izd.), Philadelphia, PA: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-425-8
- Manning, Christopher D.; Schütze, Hinrich (1999), Foundations of statistical natural language processing, MIT Press, ISBN 978-0-262-13360-9
- Mehata, K. M.; Srinivasan, S. K. (1978), Stochastic processes, New York, NY: McGraw–Hill, ISBN 978-0-07-096612-3
- Mirsky, Leonid (1990), An Introduction to Linear Algebra, Courier Dover Publications, ISBN 978-0-486-66434-7
- Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd izd.), New York: Wiley, LCCN 76-91646
- Nocedal, Jorge; Wright, Stephen J. (2006), Numerical Optimization (2nd izd.), Berlin, DE; New York, NY: Springer-Verlag, str. 449, ISBN 978-0-387-30303-1
- Oualline, Steve (2003), Practical C++ programming, O'Reilly, ISBN 978-0-596-00419-4
- Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; Vetterling, William T. (1992), „LU Decomposition and Its Applications” (PDF), Numerical Recipes in FORTRAN: The Art of Scientific Computing (2nd izd.), Cambridge University Press, str. 34—42, Arhivirano iz originala 2009-09-06. g.
- Protter, Murray H.; Morrey, Jr., Charles B. (1970), College Calculus with Analytic Geometry (2nd izd.), Reading: Addison-Wesley, LCCN 76087042
- Punnen, Abraham P.; Gutin, Gregory (2002), The traveling salesman problem and its variations, Boston, MA: Kluwer Academic Publishers, ISBN 978-1-4020-0664-7
- Reichl, Linda E. (2004), The transition to chaos: conservative classical systems and quantum manifestations, Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-98788-0
- Rowen, Louis Halle (2008), Graduate Algebra: noncommutative view, Providence, RI: American Mathematical Society, ISBN 978-0-8218-4153-2
- Šolin, Pavel (2005), Partial Differential Equations and the Finite Element Method, Wiley-Interscience, ISBN 978-0-471-76409-0
- Stinson, Douglas R. (2005), Cryptography, Discrete Mathematics and its Applications, Chapman & Hall/CRC, ISBN 978-1-58488-508-5
- Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd izd.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-95452-3
- Ward, J. P. (1997), Quaternions and Cayley numbers
, Mathematics and its Applications, 403, Dordrecht, NL: Kluwer Academic Publishers Group, ISBN 978-0-7923-4513-8, MR 1458894, doi:10.1007/978-94-011-5768-1
- Wolfram, Stephen (2003), The Mathematica Book (5th izd.), Champaign, IL: Wolfram Media, ISBN 978-1-57955-022-6
- Bohm, Arno (2001), Quantum Mechanics: Foundations and Applications, Springer, ISBN 0-387-95330-2
- Burgess, Cliff; Moore, Guy (2007), The Standard Model. A Primer, Cambridge University Press, ISBN 978-0-521-86036-9
- Guenther, Robert D. (1990), Modern Optics, John Wiley, ISBN 0-471-60538-7
- Itzykson, Claude; Zuber, Jean-Bernard (1980), Quantum Field Theory
, McGraw–Hill, ISBN 0-07-032071-3
- Riley, Kenneth F.; Hobson, Michael P.; Bence, Stephen J. (1997), Mathematical methods for physics and engineering, Cambridge University Press, ISBN 0-521-55506-X
- Schiff, Leonard I. (1968), Quantum Mechanics (3rd izd.), McGraw–Hill
- Weinberg, Steven (1995), The Quantum Theory of Fields. Volume I: Foundations, Cambridge University Press, ISBN 0-521-55001-7
- Wherrett, Brian S. (1987), Group Theory for Atoms, Molecules and Solids, Prentice–Hall International, ISBN 0-13-365461-3
- Zabrodin, Anton; Brezin, Édouard; Kazakov, Vladimir; Serban, Didina; Wiegmann, Paul (2006), Applications of Random Matrices in Physics (NATO Science Series II: Mathematics, Physics and Chemistry), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-1-4020-4530-1
- A. Cayley A memoir on the theory of matrices. Phil. Trans. 148 1858 17-37; Math. Papers II 475-496
- Bôcher, Maxime (2004), Introduction to higher algebra, New York, NY: Dover Publications, ISBN 978-0-486-49570-5, reprint of the 1907 original edition
- Cayley, Arthur (1889), The collected mathematical papers of Arthur Cayley, I (1841–1853), Cambridge University Press, str. 123—126
- Dieudonné, Jean, ur. (1978), Abrégé d'histoire des mathématiques 1700-1900, Paris, FR: Hermann
- Hawkins, Thomas (1975), „Cauchy and the spectral theory of matrices”, Historia Mathematica, 2: 1—29, ISSN 0315-0860, MR 0469635, doi:10.1016/0315-0860(75)90032-4
- Knobloch, Eberhard (1994), „From Gauss to Weierstrass: determinant theory and its historical evaluations”, The intersection of history and mathematics, Science Networks Historical Studies, 15, Basel, Boston, Berlin: Birkhäuser, str. 51—66, MR 1308079
- Kronecker, Leopold (1897), Hensel, Kurt, ur., Leopold Kronecker's Werke, Teubner
- Mehra, Jagdish; Rechenberg, Helmut (1987), The Historical Development of Quantum Theory (1st izd.), Berlin, DE; New York, NY: Springer-Verlag, ISBN 978-0-387-96284-9
- Shen, Kangshen; Crossley, John N.; Lun, Anthony Wah-Cheung (1999), Nine Chapters of the Mathematical Art, Companion and Commentary (2nd izd.), Oxford University Press, ISBN 978-0-19-853936-0
- Weierstrass, Karl (1915), Collected works, 3
- Hazewinkel, Michiel, ur. (2001) [1994], „Matrix”, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Kaw, Autar K. (septembar 2008), Introduction to Matrix Algebra, ISBN 978-0-615-25126-4
- The Matrix Cookbook (PDF), Pristupljeno 24. 3. 2014
- Brookes, Mike (2005), The Matrix Reference Manual, London: Imperial College, Pristupljeno 10. 12. 2008