Linearne enačbe. Matrična algebra. Linearne enačbe. Linearne enačbe. Linearne enačbe. Linearne enačbe
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1 Sistem linearnih enačb Matrična algebra Oseba X X X3 B A.A. 3 B.B. 7 C.C. Doc. dr. Anja Podlesek Oddelek za psihologijo, Filozofska fakulteta, Univerza v Ljubljani Študijski program prve stopnje Psihologija 3. semester, Korelacijske metode Katere vrednosti neznank rešujejo vse enačbe naenkrat? Uporaba: -obdelava digitalnih signalov, -računalniška grafika, -ocene -napovedi B = *X + 3*X + *X3 Sistem linearnih enačb neznanke koeficienti sistema linearnih enačb konstante Če so konstante 0, je sistem linearnih enačb HOMOGEN. Vsak homogeni sistem ima vsaj eno rešitev, tj. trivialno rešitev, ki jo dobimo tako, da vsem spremenljivkam pripišemo vrednost 0. V enačbah se neznanke pojavljajo samo na prvo potenco, ne pojavljajo se zmnožki. Hitro ugotovimo, ali so taki sistemi rešljivi ali ne. matrični račun: matrike determinante operacije z matrikami Matrike omogočajo enostaven zapis in reševanje sistema linearnih enačb.
2 Matrike Matrike in matrična algebra matrika reda m x n (m, n dimenzije matrike) a a a n a a a n A am am amn stolpec matrice j, < j < n element matrike ali komponenta matrike vrsta matrice i, < i < m diagonala matrike Primer: matrika reda x 3: M Kdaj so matrike enake? A in B sta enaki, če sta istega tipa in če je a ij = b ij za vse pare indeksov i, j. Matrike označujemo z velikimi tiskanimi črkami v mastnem tisku. a ij a ji Transponirana matrika Prva vrstica matrike postane prvi stolpec, druga vrstica drugi stolpec itd. 3 7 A 5 6 T A ' A B ' B Po transponiranju je red matrike obrnjen (B: x 3 B : 3 x ). VEKTORJI: Tipi matrik m = vrstična matrika (ima samo eno vrstico) n = stolpčna matrika (ima samo en stolpec) r ' s 0 Vektorje označujemo z malimi tiskanimi črkami v mastnem tisku.
3 Tipi matrik Tipi matrik Ničelna matrika ima vse elemente enake ničli Kvadratna matrika: n x n Simetrična matrika (A = A ; pr.: korelacijska matrika) Diagonalna matrika (0 izven diagonale) Spodnja in zgornja trikotna matrika 6 A D Tipi matrik Pogoste matrike v statistiki Kvadratna matrika: n x n Skalarna matrika (diagonalna z enakimi elementi v diagonali) S matrike podatkov Identična matrika (enotska) d = *0+, C = *x+ kvadratna matrika reda 0 0 I Iz: Care (998) Predpostavka: neodvisni podatki Pogoste matrike v statistiki matrike aritmetičnih sredin M M M M 3 matrike standardnih deviacij σ σ 0 S 0 0 σn Pogoste matrike v statistiki kovariančne matrike cov cov covn σ cov cov n cov cov cov n cov σ covn C covn covn covnn cov n covn σn korelacijske matrike r R rn r n n r n r r 0 0, R 0 0, 3 0, 0, 3 3
4 Sled matrike Sled (trace) kvadratne matrike je vsota diagonalnih elementov. 3 7 tr Sled matrike Koliko znaša sled v kovariančni matriki? Koliko znaša sled v korelacijski matriki? Operacije z matrikami Seštevanje matrik seštevanje množenje: s skalarjem množenje matrik transponiranje elementarne operacije na matrikah Seštevamo lahko le matrike istega tipa. Če sta matriki A in B istega tipa, je matrika C = A + B istega tipa kot A in kot B. c ij = a ij + b ij Seštevanje matrik Lastnosti seštevanja matrik komutativnost A + B = B + A asociativnost A + (B + C) = (A + B) + C
5 Množenje matrike s skalarjem (Vektorsko) Množenje matrik B ij = λa ij Lastnosti: λ A + B = λa + λb λ + μ A = λa + μa λ μa = λμ A distributivnost asociativnost Matriki A in B lahko pomnožimo samo, če sta poravnani, tj. če ima A toliko stolpcev, kot ima B vrstic. Če je A matrika reda m x k in B matrika reda k x n, je C = AB matrika reda m x n. Množenje v obratnem vrstnem redu da drugačen rezultat!!! Zmnožki nisi A x3 B 3x =C x A 3x B x3 =C 3x istega tipa (reda) Množenje matrik x ' ax b ' cx d x ' a b x x' Ax ' c d x '' x ' ' '' x ' ' x'' = Bx' x'' = B(Ax) = BAx x '' x ' a b x '' ' c d Množenje matrik x '' ( ax b) ( cx d) ( a c) x ( b d) '' ( ax b) ( cx d) ( a c) x ( b d) x '' a c b d x a c b d '' a c b d BA a c b d Množenje matrik Množenje matrik V naslednjem primeru so zmnožki istega tipa (reda), a niso enaki: Vendar pa je sled, tr(a), oz. vsota diagonalnih elementov v kvadratnih matrikah, enaka v AB in BA. A je postmultiplicirana (množena z desne) z B, oz. B je premultiplicirana (množena z leve) z A. A je premultiplicirana z B, oz. B je postmultiplicirana z A. 5
6 Množenje matrik V tem primeru pa sta oba zmnožka enaka: Lastnosti množenja matrik asociativnost: (AB)C = A(BC) distributivnost : A(B + C) = AB + AC distributivnost: (A + B)C = AC + BC l(ab) = (la)b = A(lB) Toda: Množenje matrik na splošno ni komutativno, AB BA. Identična matrika Operacije na transponiranih matrikah Pri množenju matrik je identična matrika nevtralni element = pri množenju z identično matriko se matrika A ne spremeni. Za vsako matriko tipa m x n drži: Če matrika ni kvadratna, sta enotski matriki glede na množenje z leve in z desne različnega reda. Matrika, za katero je A T = A, je simetrična matrika. Kvadratne matrike imajo enako število vrstic in stolpcev matrike istega reda lahko seštevamo in množimo kvadratna matrika A se imenuje invertibilna ali nesingularna, če obstaja matrika B, pri kateri je AB = I n ali BA = I n ; B je inverzna (obratna) matrika matrike A, tj. A - Torej: AA - = I n ali A - A = I n Kvadratne matrike imajo determinanto. Determinanta matrike Determinanta matrike je število (skalar), definirano z elementi kvadratne matrike. x matrika: x matrika: a a a b det A A ad bc c d
7 Determinanta matrike Determinanta matrike 3 x 3 matrika: a b c det M a b c a b c a b c b c a b c b c a b c b c druge matrike: naj računalnik opravi ta posel Determinanta nam pove, ali so vrstice ali stolpci matrike neodvisni. Det = 0 matrika je singularna: če stolpci/vrstice niso linearno neodvisni (z elementarnimi det operacijami lahko dobimo en stolpec/vrstico iz drugih) če so vsi elementi nekega det stolpca/vrstice enaki 0 Det = 0 inverzna matrika ne obstaja Determinanta določa (determinira), ali ima matrika inverz od tod njeno ime. Determinanta v kovariančnih in korelacijskih matrikah Za kovariančne in korelacijske matrike je determinanta število, ki ga uporabljamo, da izrazimo generalizirano varianco mora biti > 0, da je uporaba matrike smiselna; Correlation matrix is not positive semi definite = Determinanta je 0, če so neke spremenljivke v popolnosti korelirane. Kovariančne matrike, ki imajo majhne determinante, označujejo spremenljivke, ki so redundantne ali visoko korelirane. Moramo biti pozorni, ker so lahko elementi inverzne matrike nenatančno izračunani. Velike determinante matrik kažejo, da so spremenljivke med seboj neodvisne. Za multivariatne statistike so lastnosti determinante zelo pomembne (npr. v multipli regresiji lahko ΔR izrazimo kot razmerje determinant dveh modelov). Inverz matrike V skalarni algebri je inverz nekega števila X število Y, ki, če ga množimo z originalnim številom, da zmnožek (X Y = ili Y X = ). Y = /X oz. X - V matrični algebri je inverz matrike tista matrika, ki, če jo množimo z originalno matriko, da identično matriko (A A - = I ali A - A = I). Da ima matrika inverz, mora biti simetrična in mora imeti determinanto, različno od 0. V nekaterih primerih matrika nima inverza = singularna matrika. Inverz matrike AA - = A - A = I Primer za x matriko: A a det A a a a Inverz matrike za diagonalno matriko: 0 0 / / / /
8 Inverz kovariančne ali korelacijske matrike Za kovariančno in korelacijsko matriko inverz obstaja: če ima več oseb kot spremenljivk in če ima vsaka spremenljivka varianco, večjo od 0, če ni multikolinearnosti spremenljivk (in je torej det > 0). Rang matrike Rang matrike je lahko največ enak manjšemu številu med številom stolpcev in vrstic. Pri matriki 3 x je lahko največ 3. Rang matrike A je maksimalno število linearno neodvisnih vrstičnih ali stolpčnih vektorjev matrike. Rang matrike nam pove, v koliko dimenzijah se vršijo linearne transformacije. Če je rang matrike manjši od n, je njena determinanta 0 in matrika nima inverza. Rang matrike rang = (x 3 = x + x ) rang = (x 3 = 3x, x = x /) 8
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