Laʻana o nā nīnau kūkākūkā hoʻonui matrix
ʻO ka hoʻonui ʻana o ka matrix kahi manaʻo nui i ka algebra linear i hoʻohana pinepine ʻia ma nā ʻano like ʻole e like me ke kino, nā kiʻi kamepiula, a me ke aʻo ʻana i ka mīkini. Ma kēia ʻatikala, e kūkākūkā mākou i nā manaʻo nui o ka hoʻonui ʻana o ka matrix, ke "lula hoʻohui element-wise," a e hāʻawi pū i kekahi mau pilikia hoʻohālike a me kā lākou mau hoʻonā.
Nā Manaʻo Kumu o ka Hoʻonui Matrix
Ma mua o ka nānā ʻana i nā pilikia hoʻohālike, he mea nui e hoʻomaopopo i nā lula kumu o ka hoʻonui ʻana o ka matrix. Manaʻo mākou he ʻelua matrices kā mākou \( A \) a me \( B \) kahi:
– ʻO ka matrix \( A \) he nui \( m \times n \)
– ʻO ka matrix \( B \) he nui \( n \times p \)
No ka hoʻonui ʻana i ʻelua matrices \( A \) a me \( B \), pono e like ka helu o nā kolamu o ka matrice \( A \) me ka helu o nā lālani o ka matrice \( B \) (ʻo ia hoʻi nā \( n \) ʻelua). ʻO ka huahana o kēia mau matrices he matrice \( C \) o ka nui \( m \times p \) kahi i wehewehe ʻia ai nā mea \( C_{ij} \) penei:
\[ C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj} \]
ʻO ke ʻano kēia, ʻo kēlā me kēia element o ka matrix hopena ka huina o nā huahana o nā element o ka lālani \(i \) o ka matrix \(A \) me nā element o ke kolamu \(j \) o ka matrix \(B \).
Nā Nīnau Laʻana a me ke Kūkākūkā
Nīnau 1: Hoʻonui ʻia o nā Matrices 2 × 2
Manaʻo ʻia he mau matrices \( A \) a me \( B \) kā mākou e like me kēia:
\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \]
\[ B = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix} \]
E hoʻonui i nā matrices \( A \) a me \( B \) e loaʻa ai ka matrix hopena \( C \).
Kūkākūkā:
E helu kākou i nā ʻāpana o ka matrix \( C \):
\[ C_{11} = 1 \cdot 2 + 2 \cdot 1 = 2 + 2 = 4 \]
\[ C_{12} = 1 \cdot 0 + 2 \cdot 3 = 0 + 6 = 6 \]
\[ C_{21} = 3 \cdot 2 + 4 \cdot 1 = 6 + 4 = 10 \]
\[ C_{22} = 3 \cdot 0 + 4 \cdot 3 = 0 + 12 = 12 \]
No laila, ʻo ka matrix hopena \( C \) penei:
\[ C = \begin{pmatrix} 4 & 6 \\ 10 & 12 \end{pmatrix} \]
Nīnau 2: Hoʻonui ʻia o nā Matrices 3 × 3
Manaʻo ʻia he mau matrices \( D \) a me \( E \) kā mākou e like me kēia:
\[ D = \begin{pmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 0 \end{pmatrix} \]
\[ E = \begin{pmatrix} 3 & 1 & 2 \\ 2 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix} \]
E hoʻonui i nā matrices \( D \) a me \( E \) e loaʻa ai ka matrix hopena \( F \).
Kūkākūkā:
E helu kākou i nā ʻāpana o ka matrix \( F \):
\[ F_{11} = 1 \cdot 3 + 0 \cdot 2 + 2 \cdot 1 = 3 + 0 + 2 = 5 \]
\[ F_{12} = 1 \cdot 1 + 0 \cdot 1 + 2 \cdot 0 = 1 + 0 + 0 = 1 \]
\[ F_{13} = 1 \cdot 2 + 0 \cdot 1 + 2 \cdot 1 = 2 + 0 + 2 = 4 \]
\[ F_{21} = -1 \cdot 3 + 3 \cdot 2 + 1 \cdot 1 = -3 + 6 + 1 = 4 \]
\[ F_{22} = -1 \cdot 1 + 3 \cdot 1 + 1 \cdot 0 = -1 + 3 + 0 = 2 \]
\[ F_{23} = -1 \cdot 2 + 3 \cdot 1 + 1 \cdot 1 = -2 + 3 + 1 = 2 \]
\[ F_{31} = 2 \cdot 3 + 1 \cdot 2 + 0 \cdot 1 = 6 + 2 + 0 = 8 \]
\[ F_{32} = 2 \cdot 1 + 1 \cdot 1 + 0 \cdot 0 = 2 + 1 + 0 = 3 \]
\[ F_{33} = 2 \cdot 2 + 1 \cdot 1 + 0 \cdot 1 = 4 + 1 + 0 = 5 \]
No laila, ʻo ka matrix hopena \( F \) penei:
\[ F = \begin{pmatrix} 5 & 1 & 4 \\ 4 & 2 & 2 \\ 8 & 3 & 5 \end{pmatrix} \]
Nīnau 3: Hoʻonui ʻia o kahi Matrix 2 × 3 e kahi Matrix 3 × 2
Manaʻo ʻia he mau matrices \( G \) a me \( H \) kā mākou e like me kēia:
\[ G = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \]
\[ H = \begin{pmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{pmatrix} \]
E hoʻonui i nā matrices \( G \) a me \( H \) e loaʻa ai ka matrix hopena \( I \).
Kūkākūkā:
E helu kākou i nā ʻāpana o ka matrix \( I \):
\[ I_{11} = 1 \cdot 7 + 2 \cdot 9 + 3 \cdot 11 = 7 + 18 + 33 = 58 \]
\[ I_{12} = 1 \cdot 8 + 2 \cdot 10 + 3 \cdot 12 = 8 + 20 + 36 = 64 \]
\[ I_{21} = 4 \cdot 7 + 5 \cdot 9 + 6 \cdot 11 = 28 + 45 + 66 = 139 \]
\[ I_{22} = 4 \cdot 8 + 5 \cdot 10 + 6 \cdot 12 = 32 + 50 + 72 = 154 \]
No laila, ʻo ka matrix hopena \(I \) penei:
\[ I = \begin{pmatrix} 58 & 64 \\ 139 & 154 \end{pmatrix} \]
Ka hopena
Ma kēia ʻatikala, ua uhi mākou i nā lula kumu o ka hoʻonui matrix a ua hāʻawi i ʻekolu mau pilikia hoʻohālike me nā wehewehe. He ʻōnaehana ke kaʻina hana o ka helu ʻana i ka hoʻonui matrix, e koi ana i ka nānā kikoʻī i nā mea hoʻonui o kēlā me kēia mea matrix a me kā lākou huina. Ma ka hoʻomaopopo ʻana a me ka hoʻomaʻamaʻa pinepine ʻana i nā pilikia hoʻonui matrix, e hoʻomaopopo maikaʻi mākou i kēia manaʻo a hiki ke hoʻopili iā ia ma nā ʻano ʻepekema like ʻole.
ʻAʻole wale ka hoʻonui ʻana o ka matrix he kahua koʻikoʻi i ka makemakika a me ka ʻepekema kamepiula, akā he mea pono loa hoʻi i nā noi o ke ao maoli, e like me ka nānā ʻana i ka ʻikepili, ka hoʻonui ʻana, a me nā algorithms aʻo mīkini. No laila, ʻo ka hoʻomaopopo maikaʻi ʻana i ka hoʻonui ʻana o ka matrix he kahua koʻikoʻi ia no kekahi makemakika a i ʻole ka ʻepekema kamepiula.
