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.

Definicije i notacije uredi

Horizontalne linije u matrici se nazivaju vrstama, a vertikalne kolonama matrice.^[1]

Preslikavanje $A:\{1,2,...,m\}\times \{1,2,...,n\}\rightarrow F$ , takvo da je $(F,+,\cdot )$ polje i $m,n\in \mathbb {N}$ nazivamo matricom tipa $m\times n$ 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 A_i,j ili A[i,j]. Uvek se prvo naznačuje vrsta, pa kolona.

Često se piše $A:=(a_{i,j})_{m\times n}$ kako bi se definisala m × n matrica A čiji se svaki član, A[i,j] naziva a_i,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

Matrica

A={\begin{bmatrix}1&2&3\\1&2&7\\4&9&2\\6&0&5\end{bmatrix}}

je 4×3 matrica. Element A[2,3] ili a_2,3 je 7.

Matrica

R={\begin{bmatrix}1&2&3&4&5&6&7&8&9\end{bmatrix}}

je 1×9 matrica, ili vektor vrsta sa 9 elemenata.

Sabiranje i množenje matrica uredi

Neka su date matrice $A=[a_{i,j}]_{m\times n}$ i $B=[b_{i,j}]_{m\times n}$ .

Sabiranje uredi

Zbir matrica A i V, u oznaci A+V je matrica $C=[c_{i,j}]_{m\times n}$ za koju važi $c_{i,j}=a_{i,j}+b_{i,j}$ za svako $i=\{1,2,...,m\};j=\{1,2,...,n\}$ .

{\begin{bmatrix}1&3&2\\1&0&0\\1&2&2\end{bmatrix}}+{\begin{bmatrix}0&0&5\\7&5&0\\2&1&1\end{bmatrix}}={\begin{bmatrix}1+0&3+0&2+5\\1+7&0+5&0+0\\1+2&2+1&2+1\end{bmatrix}}={\begin{bmatrix}1&3&7\\8&5&0\\3&3&3\end{bmatrix}}

Množenje skalarom uredi

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:

2{\begin{bmatrix}1&8&-3\\4&-2&5\end{bmatrix}}={\begin{bmatrix}2\cdot 1&2\cdot 8&2\cdot (-3)\\2\cdot 4&2\cdot (-2)&2\cdot 5\end{bmatrix}}={\begin{bmatrix}2&16&-6\\8&-4&10\end{bmatrix}}

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

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:

\,\!(AB)[i,j]=A[i,1]B[1,j]+A[i,2]B[2,j]+...+A[i,n]B[n,j]

za svaki par i i j.

Na primer:

{\begin{bmatrix}1&0&2\\-1&3&1\\\end{bmatrix}}\cdot {\begin{bmatrix}3&1\\2&1\\1&0\\\end{bmatrix}}={\begin{bmatrix}(1\cdot 3+0\cdot 2+2\cdot 1)&(1\cdot 1+0\cdot 1+2\cdot 0)\\((-1)\cdot 3+3\cdot 2+1\cdot 1)&((-1)\cdot 1+3\cdot 1+1\cdot 0)\\\end{bmatrix}}

={\begin{bmatrix}5&1\\4&2\\\end{bmatrix}}

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

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 → R^m 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 : R^m → R^k, tada je njihova kompozicija g o f takođe linearno preslikavanje R^m → 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 A^tr (nekad se zapisuje i kao A^T ili ^tA), koja nastaje pretvaranjem vrsta u kolone, i kolona u vrste, to jest A^tr[i, j] = A[j, i] za svako i i j. Ako A predstavlja linearno preslikavanje u odnosu na dve baze, tada matrica A^tr predstavlja linearno preslikavanje u odnosu na dualne baze (vidi dualni prostor).

Važi (A + B)^tr = A^tr + B^tr i (AB)^tr = B^tr A^tr.

Vidi još uredi

Osobine matrica uredi

Posebne matrice uredi

Reference uredi

^ Lang 2002

Literatura uredi

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. CS1 održavanje: Nepodoban URL (veza)
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 CS1 održavanje: Format datuma (veza)
The Matrix Cookbook (PDF), Pristupljeno 24. 3. 2014 CS1 održavanje: Format datuma (veza)
Brookes, Mike (2005), The Matrix Reference Manual, London: Imperial College, Pristupljeno 10. 12. 2008 CS1 održavanje: Format datuma (veza)

Spoljašnje veze uredi

Matrice
MacTutor: Matrices and determinants Arhivirano na sajtu Wayback Machine (8. mart 2015)
Matrices and Linear Algebra on the Earliest Uses Pages
Earliest Uses of Symbols for Matrices and Vectors

[1] Lang 2002

[1]