Matricu reizināšana ir bināra operācija , kurā tiek reizināts matricu pāris, šīs operācijas rezultātā tiek iegūta jauna matrica. Skaitļi (piemēram, reāli , kompleksi skaitļi) var tikt reizināti kā elementārajā aritmētikā.
Pastāv vairāki veidi, kā reizināt matricas, no kuriem vienkāršākais ir reizināšana ar skaitli. Matricu reizināšana nav komutatīva , kas nozīmē, ka A · B ≠ B · A A n × m B m × p AB n × p
Reizināšana
Reizināšana ar skaitli Vienkāršākā reizināšana ar matricām ir matricas reizināšana ar skaitli.
Skaitļa λ A λ A A λ A
( λ A ) i j = λ ( A ) i j , {\displaystyle (\lambda \mathbf {A} )_{ij}=\lambda \left(\mathbf {A} \right)_{ij}\,,} kas izvērstā pierakstā izskatās šādi:
λ A = λ ( A 11 A 12 ⋯ A 1 m A 21 A 22 ⋯ A 2 m ⋮ ⋮ ⋱ ⋮ A n 1 A n 2 ⋯ A n m ) = ( λ A 11 λ A 12 ⋯ λ A 1 m λ A 21 λ A 22 ⋯ λ A 2 m ⋮ ⋮ ⋱ ⋮ λ A n 1 λ A n 2 ⋯ λ A n m ) . {\displaystyle \lambda \mathbf {A} =\lambda {\begin{pmatrix}A_{11}&A_{12}&\cdots &A_{1m}\\A_{21}&A_{22}&\cdots &A_{2m}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nm}\\\end{pmatrix}}={\begin{pmatrix}\lambda A_{11}&\lambda A_{12}&\cdots &\lambda A_{1m}\\\lambda A_{21}&\lambda A_{22}&\cdots &\lambda A_{2m}\\\vdots &\vdots &\ddots &\vdots \\\lambda A_{n1}&\lambda A_{n2}&\cdots &\lambda A_{nm}\\\end{pmatrix}}\,.} Līdzīgi tiek definēts matricas A λ
( A λ ) i j = ( A ) i j λ , {\displaystyle (\mathbf {A} \lambda )_{ij}=\left(\mathbf {A} \right)_{ij}\lambda \,,} kas izvērstā pierakstā izskatās šādi:
A λ = ( A 11 A 12 ⋯ A 1 m A 21 A 22 ⋯ A 2 m ⋮ ⋮ ⋱ ⋮ A n 1 A n 2 ⋯ A n m ) λ = ( A 11 λ A 12 λ ⋯ A 1 m λ A 21 λ A 22 λ ⋯ A 2 m λ ⋮ ⋮ ⋱ ⋮ A n 1 λ A n 2 λ ⋯ A n m λ ) . {\displaystyle \mathbf {A} \lambda ={\begin{pmatrix}A_{11}&A_{12}&\cdots &A_{1m}\\A_{21}&A_{22}&\cdots &A_{2m}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nm}\\\end{pmatrix}}\lambda ={\begin{pmatrix}A_{11}\lambda &A_{12}\lambda &\cdots &A_{1m}\lambda \\A_{21}\lambda &A_{22}\lambda &\cdots &A_{2m}\lambda \\\vdots &\vdots &\ddots &\vdots \\A_{n1}\lambda &A_{n2}\lambda &\cdots &A_{nm}\lambda \\\end{pmatrix}}\,.} Piemērs ar reālu skaitli un matricu:
λ = 2 , A = ( a b c d ) {\displaystyle \lambda =2,\quad \mathbf {A} ={\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}} 2 A = 2 ( a b c d ) = ( 2 ⋅ a 2 ⋅ b 2 ⋅ c 2 ⋅ d ) = ( a ⋅ 2 b ⋅ 2 c ⋅ 2 d ⋅ 2 ) = ( a b c d ) 2 = A 2. {\displaystyle 2\mathbf {A} =2{\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}={\begin{pmatrix}2\!\cdot \!a&2\!\cdot \!b\\2\!\cdot \!c&2\!\cdot \!d\\\end{pmatrix}}={\begin{pmatrix}a\!\cdot \!2&b\!\cdot \!2\\c\!\cdot \!2&d\!\cdot \!2\\\end{pmatrix}}={\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}2=\mathbf {A} 2.}
Divu matricu reizināšana Ja A n × m B m × p
A = ( A 11 A 12 ⋯ A 1 m A 21 A 22 ⋯ A 2 m ⋮ ⋮ ⋱ ⋮ A n 1 A n 2 ⋯ A n m ) , B = ( B 11 B 12 ⋯ B 1 p B 21 B 22 ⋯ B 2 p ⋮ ⋮ ⋱ ⋮ B m 1 B m 2 ⋯ B m p ) {\displaystyle \mathbf {A} ={\begin{pmatrix}A_{11}&A_{12}&\cdots &A_{1m}\\A_{21}&A_{22}&\cdots &A_{2m}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nm}\\\end{pmatrix}},\quad \mathbf {B} ={\begin{pmatrix}B_{11}&B_{12}&\cdots &B_{1p}\\B_{21}&B_{22}&\cdots &B_{2p}\\\vdots &\vdots &\ddots &\vdots \\B_{m1}&B_{m2}&\cdots &B_{mp}\\\end{pmatrix}}} tad matricu reizinājums AB n × p [1] [2] [3] [4]
A B = ( ( A B ) 11 ( A B ) 12 ⋯ ( A B ) 1 p ( A B ) 21 ( A B ) 22 ⋯ ( A B ) 2 p ⋮ ⋮ ⋱ ⋮ ( A B ) n 1 ( A B ) n 2 ⋯ ( A B ) n p ) {\displaystyle \mathbf {A} \mathbf {B} ={\begin{pmatrix}\left(\mathbf {AB} \right)_{11}&\left(\mathbf {AB} \right)_{12}&\cdots &\left(\mathbf {AB} \right)_{1p}\\\left(\mathbf {AB} \right)_{21}&\left(\mathbf {AB} \right)_{22}&\cdots &\left(\mathbf {AB} \right)_{2p}\\\vdots &\vdots &\ddots &\vdots \\\left(\mathbf {AB} \right)_{n1}&\left(\mathbf {AB} \right)_{n2}&\cdots &\left(\mathbf {AB} \right)_{np}\\\end{pmatrix}}} kur katrs i, j Aik Bkj k = 1, 2, ..., m k
( A B ) i j = ∑ k = 1 m A i k B k j . {\displaystyle (\mathbf {A} \mathbf {B} )_{ij}=\sum _{k=1}^{m}A_{ik}B_{kj}\,.} Tas nozīmē, ka reizinājums AB A B A B
Ilustrācija [ a 11 a 12 ⋅ ⋅ a 31 a 32 ⋅ ⋅ ] 4 × 2 matrica [ ⋅ b 12 b 13 ⋅ b 22 b 23 ] 2 × 3 matrica = [ ⋅ x 12 x 13 ⋅ ⋅ ⋅ ⋅ x 32 x 33 ⋅ ⋅ ⋅ ] 4 × 3 matrica {\displaystyle {\overset {4\times 2{\text{ matrica}}}{\begin{bmatrix}{\color {Brown}{a_{11}}}&{\color {Brown}{a_{12}}}\\\cdot &\cdot \\{\color {Orange}{a_{31}}}&{\color {Orange}{a_{32}}}\\\cdot &\cdot \\\end{bmatrix}}}{\overset {2\times 3{\text{ matrica}}}{\begin{bmatrix}\cdot &{\color {Plum}{b_{12}}}&{\color {Violet}{b_{13}}}\\\cdot &{\color {Plum}{b_{22}}}&{\color {Violet}{b_{23}}}\\\end{bmatrix}}}={\overset {4\times 3{\text{ matrica}}}{\begin{bmatrix}\cdot &x_{12}&x_{13}\\\cdot &\cdot &\cdot \\\cdot &x_{32}&x_{33}\\\cdot &\cdot &\cdot \\\end{bmatrix}}}} Vērtības krustpunktos ir iegūstamas ar šādām darbībām:
x 12 = a 11 b 12 + a 12 b 22 x 13 = a 11 b 13 + a 12 b 23 x 32 = a 31 b 12 + a 32 b 22 x 33 = a 31 b 13 + a 32 b 23 {\displaystyle {\begin{aligned}x_{12}&={\color {Brown}{a_{11}}}{\color {Plum}{b_{12}}}+{\color {Brown}{a_{12}}}{\color {Plum}{b_{22}}}\\x_{13}&={\color {Brown}{a_{11}}}{\color {Violet}{b_{13}}}+{\color {Brown}{a_{12}}}{\color {Violet}{b_{23}}}\\x_{32}&={\color {Orange}{a_{31}}}{\color {Plum}{b_{12}}}+{\color {Orange}{a_{32}}}{\color {Plum}{b_{22}}}\\x_{33}&={\color {Orange}{a_{31}}}{\color {Violet}{b_{13}}}+{\color {Orange}{a_{32}}}{\color {Violet}{b_{23}}}\end{aligned}}}
Matricu reizināšanas piemēri Rindas matrica un kolonnas matrica Ja
A = ( a b c ) , B = ( x y z ) , {\displaystyle \mathbf {A} ={\begin{pmatrix}a&b&c\end{pmatrix}}\,,\quad \mathbf {B} ={\begin{pmatrix}x\\y\\z\end{pmatrix}}\,,} tad matricu reizināšanas rezultāts ir:
A B = ( a b c ) ( x y z ) = a x + b y + c z , {\displaystyle \mathbf {AB} ={\begin{pmatrix}a&b&c\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}=ax+by+cz\,,} un
B A = ( x y z ) ( a b c ) = ( x a x b x c y a y b y c z a z b z c ) . {\displaystyle \mathbf {BA} ={\begin{pmatrix}x\\y\\z\end{pmatrix}}{\begin{pmatrix}a&b&c\end{pmatrix}}={\begin{pmatrix}xa&xb&xc\\ya&yb&yc\\za&zb&zc\end{pmatrix}}\,.} Jāievēro, ka AB BA 1 × 1 matrica, bet otrā — 3 × 3 matrica.
Kvadrātiska matrica un kolonnas matrica Ja
A = ( a b c p q r u v w ) , B = ( x y z ) , {\displaystyle \mathbf {A} ={\begin{pmatrix}a&b&c\\p&q&r\\u&v&w\end{pmatrix}},\quad \mathbf {B} ={\begin{pmatrix}x\\y\\z\end{pmatrix}}\,,} tad matricu reizināšanas rezultāts ir:
A B = ( a b c p q r u v w ) ( x y z ) = ( a x + b y + c z p x + q y + r z u x + v y + w z ) , {\displaystyle \mathbf {AB} ={\begin{pmatrix}a&b&c\\p&q&r\\u&v&w\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}ax+by+cz\\px+qy+rz\\ux+vy+wz\end{pmatrix}}\,,} savukārt BA
Kvadrātiskas matricas Ja
A = ( a b c p q r u v w ) , B = ( α β γ λ μ ν ρ σ τ ) , {\displaystyle \mathbf {A} ={\begin{pmatrix}a&b&c\\p&q&r\\u&v&w\end{pmatrix}},\quad \mathbf {B} ={\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}\,,} tad matricu reizināšanas rezultāts ir:
A B = ( a b c p q r u v w ) ( α β γ λ μ ν ρ σ τ ) = ( a α + b λ + c ρ a β + b μ + c σ a γ + b ν + c τ p α + q λ + r ρ p β + q μ + r σ p γ + q ν + r τ u α + v λ + w ρ u β + v μ + w σ u γ + v ν + w τ ) , {\displaystyle \mathbf {AB} ={\begin{pmatrix}a&b&c\\p&q&r\\u&v&w\end{pmatrix}}{\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}={\begin{pmatrix}a\alpha +b\lambda +c\rho &a\beta +b\mu +c\sigma &a\gamma +b\nu +c\tau \\p\alpha +q\lambda +r\rho &p\beta +q\mu +r\sigma &p\gamma +q\nu +r\tau \\u\alpha +v\lambda +w\rho &u\beta +v\mu +w\sigma &u\gamma +v\nu +w\tau \end{pmatrix}}\,,} un
B A = ( α β γ λ μ ν ρ σ τ ) ( a b c p q r u v w ) = ( α a + β p + γ u α b + β q + γ v α c + β r + γ w λ a + μ p + ν u λ b + μ q + ν v λ c + μ r + ν w ρ a + σ p + τ u ρ b + σ q + τ v ρ c + σ r + τ w ) . {\displaystyle \mathbf {BA} ={\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}{\begin{pmatrix}a&b&c\\p&q&r\\u&v&w\end{pmatrix}}={\begin{pmatrix}\alpha a+\beta p+\gamma u&\alpha b+\beta q+\gamma v&\alpha c+\beta r+\gamma w\\\lambda a+\mu p+\nu u&\lambda b+\mu q+\nu v&\lambda c+\mu r+\nu w\\\rho a+\sigma p+\tau u&\rho b+\sigma q+\tau v&\rho c+\sigma r+\tau w\end{pmatrix}}\,.} Rindas matrica, kvadrātiska matrica un kolonnas matrica Ja
A = ( a b c ) , B = ( α β γ λ μ ν ρ σ τ ) , C = ( x y z ) , {\displaystyle \mathbf {A} ={\begin{pmatrix}a&b&c\end{pmatrix}}\,,\quad \mathbf {B} ={\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}\,,\quad \mathbf {C} ={\begin{pmatrix}x\\y\\z\end{pmatrix}}\,,} tad matricu reizināšanas rezultāts ir:
A B C = ( a b c ) [ ( α β γ λ μ ν ρ σ τ ) ( x y z ) ] = [ ( a b c ) ( α β γ λ μ ν ρ σ τ ) ] ( x y z ) = ( a b c ) ( α x + β y + γ z λ x + μ y + ν z ρ x + σ y + τ z ) = ( a α + b λ + c ρ a β + b μ + c σ a γ + b ν + c τ ) ( x y z ) = a α x + b λ x + c ρ x + a β y + b μ y + c σ y + a γ z + b ν z + c τ z , {\displaystyle {\begin{aligned}\mathbf {ABC} &={\begin{pmatrix}a&b&c\end{pmatrix}}\left[{\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}\right]=\left[{\begin{pmatrix}a&b&c\end{pmatrix}}{\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}\right]{\begin{pmatrix}x\\y\\z\end{pmatrix}}\\&={\begin{pmatrix}a&b&c\end{pmatrix}}{\begin{pmatrix}\alpha x+\beta y+\gamma z\\\lambda x+\mu y+\nu z\\\rho x+\sigma y+\tau z\\\end{pmatrix}}={\begin{pmatrix}a\alpha +b\lambda +c\rho &a\beta +b\mu +c\sigma &a\gamma +b\nu +c\tau \end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}\\&=a\alpha x+b\lambda x+c\rho x+a\beta y+b\mu y+c\sigma y+a\gamma z+b\nu z+c\tau z\,,\end{aligned}}} CBA A (BC ) = (AB )C
Taisnstūrveida matricas Ja
A = ( a b c x y z ) , B = ( α ρ β σ γ τ ) , {\displaystyle \mathbf {A} ={\begin{pmatrix}a&b&c\\x&y&z\end{pmatrix}}\,,\quad \mathbf {B} ={\begin{pmatrix}\alpha &\rho \\\beta &\sigma \\\gamma &\tau \\\end{pmatrix}}\,,} tad matricu reizināšanas rezultāts ir:
A B = ( a b c x y z ) ( α ρ β σ γ τ ) = ( a α + b β + c γ a ρ + b σ + c τ x α + y β + z γ x ρ + y σ + z τ ) , {\displaystyle \mathbf {A} \mathbf {B} ={\begin{pmatrix}a&b&c\\x&y&z\end{pmatrix}}{\begin{pmatrix}\alpha &\rho \\\beta &\sigma \\\gamma &\tau \\\end{pmatrix}}={\begin{pmatrix}a\alpha +b\beta +c\gamma &a\rho +b\sigma +c\tau \\x\alpha +y\beta +z\gamma &x\rho +y\sigma +z\tau \\\end{pmatrix}}\,,} un
B A = ( α ρ β σ γ τ ) ( a b c x y z ) = ( α a + ρ x α b + ρ y α c + ρ z β a + σ x β b + σ y β c + σ z γ a + τ x γ b + τ y γ c + τ z ) . {\displaystyle \mathbf {B} \mathbf {A} ={\begin{pmatrix}\alpha &\rho \\\beta &\sigma \\\gamma &\tau \\\end{pmatrix}}{\begin{pmatrix}a&b&c\\x&y&z\end{pmatrix}}={\begin{pmatrix}\alpha a+\rho x&\alpha b+\rho y&\alpha c+\rho z\\\beta a+\sigma x&\beta b+\sigma y&\beta c+\sigma z\\\gamma a+\tau x&\gamma b+\tau y&\gamma c+\tau z\end{pmatrix}}\,.}
Komutativitāte
Atsauces
Ārējās saites 
![{\displaystyle {\begin{aligned}\mathbf {ABC} &={\begin{pmatrix}a&b&c\end{pmatrix}}\left[{\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}\right]=\left[{\begin{pmatrix}a&b&c\end{pmatrix}}{\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}\right]{\begin{pmatrix}x\\y\\z\end{pmatrix}}\\&={\begin{pmatrix}a&b&c\end{pmatrix}}{\begin{pmatrix}\alpha x+\beta y+\gamma z\\\lambda x+\mu y+\nu z\\\rho x+\sigma y+\tau z\\\end{pmatrix}}={\begin{pmatrix}a\alpha +b\lambda +c\rho &a\beta +b\mu +c\sigma &a\gamma +b\nu +c\tau \end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}\\&=a\alpha x+b\lambda x+c\rho x+a\beta y+b\mu y+c\sigma y+a\gamma z+b\nu z+c\tau z\,,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43c6d62200e683fe070d5e66e7e94d11ffdfad2d)
