Sebopeho sa Matrix e otlolohileng
Li-matrice ke e 'ngoe ea likhopolo tsa bohlokoa ka ho fetisisa lipalo, haholo-holo algebra e otlolohileng. Mafapheng a fapaneng—ho tloha fisiks le lipalo-palo ho ea moruong ho ea saenseng ea lik'homphieutha—li-matrice li sebelisoa ho emela data, litsamaiso tsa li-equation, liphetoho, le tse ling tse ngata. Har'a mefuta e mengata e tsebahalang ea li-matrice, li-matrice tse otlolohileng li na le sebaka se ikhethang ka lebaka la bonolo ba tsona, leha ho le joalo matla a tsona lipalo le tlhahlobo. Sengoloa sena se tšohla tlhaloso, litšobotsi, sebopeho se akaretsang, thepa le mehlala ea li-matrice tse otlolohileng.
Ho Utloisisa Matrix e Tšoaeang
Matrix e otlolohileng ke matrix e sekwere (palo ea mela e lekana le palo ea likholomo) moo likarolo tsohle tse ka ntle ho daegonale e kholo e leng lefela. Daegonale e kholo e na le likarolo tse behiloeng ho tloha holimo ka letsohong le letšehali ho ea tlase ka ho le letona, e leng likarolo tse maemong \((1,1), (2,2), (3,3)\), jj.
Ka mantsoe a mang, ke dielemente tse ka hara daegonale e kgolo feela tse ka bang lefela, ha dielemente tse ka ntle ho daegonale e kgolo di lokela ho ba lefela. Boleng ba daegonale e kgolo e ka ba lefela kapa e seng lefela, ho itshetlehile ka nyewe.
Mohlala, matrix e latelang ke matrix e otlolohileng:
\[
\begin{pmatrix}
4 & 0 & 0 \\
0 le -2 le 0 \\
0 & 0 le 7
\end{pmatrix}
\]
Hlokomela hore likarolo tsohle tse ling ntle le 4, -2, le 7 ke lefela, kahoo matrix e khotsofatsa tlhaloso ea matrix e otlolohileng.
Sebopeho se Akaretsang sa Matrix e Tletseng
Ka kakaretso, matrix e otlolohileng ea tatellano \(n \times n\) e ka ngoloa ka tsela ena:
\[
D=
\begin{pmatrix}
d_1 le 0 le 0 le \cdots le 0 \\
0 & d_2 & 0 & \cdots & 0 \\
0 le 0 le d_3 le \cdots le 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 le 0 le 0 le \cdots le d_n
\end{pmatrix}
\]
Mona, \(d_1, d_2, \ldots, d_n\) ke dielemente tsa daegonale e kgolo. E nngwe le e nngwe ya tsona e ka ba ya nnete, ya kakaretso, kapa ya ba e rarahaneng, ho latela moelelo.
Mongolo o kgutshwane o boetse o sebediswa hangata:
\[
D = \text{diag}(d_1, d_2, \ldots, d_n)
\]
Mongolo ona o bolela hore matrix \(D\) e na le dielemente tse kgolo tse otlolohileng \(d_1\) ho ya ho \(d_n\) mme dielemente tse ding kaofela ke lefela.
Litšobotsi tsa Matrix e Tšoaeang
Litšobotsi tse ling tse etsang hore ho be bonolo ho lemoha matrices e otlolohileng ke:
1. Matrix e hlokahalang ea sekwere
Matrix e otlolohileng kamehla e boholo ba \(n \times n\), e ke ke ea ba khutlonnetsepa.
2. Dielemente tse seng tsa daegonale di lokela ho ba lefela
Dielemente tsohle \(a_{ij}\) tse nang le \(i \neq j\) di lokela ho ba 0.
3. Likarolo tse sa lefelloeng tse otlolohileng
Dielemente tse tabehileng \(a_{ii}\) e ka ba boleng bofe kapa bofe (ho kenyeletsoa le 0).
4. Matrix e otlolohileng ke mohlala o ikhethang oa matrix e khutlotharo.
Matrix e otlolohileng ke matrix e kaholimo e kgutlotharo le e ka tlase e kgutlotharo.
Kamano le Matrix ea Boitsebiso le Matrix ea Scalar
Matrices e otlolohileng e na le kamano e haufi le mefuta e meng e 'meli ea matrices e atisang ho hlaha, e leng:
1. Setšoantšo sa Boitsebiso
Matrix ea boitsebiso ke matrix e otlolohileng e nang le likarolo tsohle tse otlolohileng tse lekanang le 1:
\[
ke =
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 le 1
\end{pmatrix}
\]
Matrix ena e bohlokoa hobane e sebetsa joalo ka nomoro ea 1 ha ho atisoa: ho atisa matrix e 'ngoe ka matrix ea boitsebiso ha ho fetole matrix (ea boholo bo loketseng).
2. Scalar Matrix
Matrix ea scalar ke matrix e otlolohileng e nang le likarolo tsohle tse otlolohileng tse nang le boleng bo tšoanang, mohlala \(k\):
\[
kI =
\begin{pmatrix}
k & 0 & 0 \\
0 & k & 0 \\
0 le 0 le k
\end{pmatrix}
\]
Ka mantsoe a mang, matrix ea scalar ke mofuta o ikhethileng oa matrix e otlolohileng, 'me matrix ea boitsebiso ke mofuta o khethehileng oa matrix ea scalar.
Matlotlo a Bohlokoa a Matrices a Diagonal
Bonolo ba sebopeho sa matrix e otlolohileng bo e fa thepa e etsang hore dipalo di be bonolo haholo.
1. Ho eketsa le ho tlosa
Haeba \(D_1\) le \(D_2\) e le matrices e otlolohileng ya boholo bo tshwanang, jwale:
– \(D_1 + D_2\) le yona ke matrix e otlolohileng
– \(D_1 – D_2\) hape ke matrix e otlolohileng
Hobane ho eketsa ho etsahala feela hodima dielemente tse tsamaellanang, mme dielemente tsohle tse seng tsa daegonale di dula di le lefela.
2. Katiso ea Matrix e Tala
Sehlahisoa sa matrices tse peli tse otlolohileng le sona ke matrices e otlolohileng. Haeba:
\[
D_1 = \text{diag}(a_1, a_2, \ldots, a_n), \quad
D_2 = \text{diag}(b_1, b_2, \ldots, b_n)
\]
Kahoo:
\[
D_1D_2 = \text{diag}(a_1b_1, a_2b_2, \ldots, a_nb_n)
\]
Sena se sebetsa hantle haholo hobane ha se hloke ho etsa katiso e felletseng ya matrix e leng ntho e atisang ho ba thata.
3. Ntho e khethollang
Ho bonolo haholo ho bala ntlha e khethollang matrix e otlolohileng, e leng sehlahisoa sa likarolo tsa eona tse otlolohileng:
\[
\det(D) = d_1 \cdot d_2 \cdot \ldots \cdot d_n
\]
4. Ka lehlakoreng le leng
Matrix e otlolohileng e ka fetoloa habonolo, ha feela likarolo tsohle tse otlolohileng e se lefela.
\[
D^{-1} = \text{diag}\left(\frac{1}{d_1}, \frac{1}{d_2}, \ldots, \frac{1}{d_n}\right)
\]
Haeba karolo efe kapa efe e otlolohileng e le lefela, joale sesupa-tsela ke lefela 'me matrix ha e na se fapaneng.
5. Boemo ba Matrix
Li-exponents tsa matrix e otlolohileng le tsona li bonolo:
\[
D^k = \text{diag}(d_1^k, d_2^k, \ldots, d_n^k)
\]
Sena se thusa haholo ho baleng mehlala e fetohang le liphetoho tse ipheta-phetang.
Mehlala ea Matrices e Tletseng le e seng ea Tletseng
Mohlala oa matrix e otlolohileng:
\[
\begin{pmatrix}
3 le 0 \\
0 & 5
\end{pmatrix}
\]
Mehlala ea matrices e seng diagonal (hobane ho na le likarolo tse seng diagonal tse seng tsa lefela):
\[
\begin{pmatrix}
3 le 1 \\
0 & 5
\end{pmatrix}
\]
Leha matrix e le khutlotharo e ka hodimo, ha se matrix e otlolohileng hobane karolo (1,2) ke 1, eseng 0.
Diagonalization: Ho Fetola Matrix ho ba Sebopeho sa Diagonal
Ntle le "matrix e otlolohileng" e le mofuta oa matrix, ho na le mohopolo oa bohlokoa o bitsoang diagonalization, e leng ts'ebetso ea ho fetola matrix e fanoeng hore e be sebopeho se otlolohileng ka phetoho:
\[
A = PDP^{-1}
\]
moo \(D\) e leng matrix e otlolohileng e nang le boleng ba eigen, mme \(P\) e le matrix eo dikholomo tsa yona e leng di-eigenvector. Haeba matrix e ka arolwa ka diagonal, dipalo tse ngata tse kang ho bala maemo a matrix di ba bonolo haholo hobane ho lekane ho sebetsa le \(D\).
Saenseng le boenjiniere, ho arohanya di-diagonal hangata ho sebediswa ho rarolla ditsamaiso tse fapaneng, tlhahlobo ya botsitso, kgatello ya data, le ts'ebetso ya matshwao.
Litšebeliso tsa Diagonal Matrix Bophelong ba Sebele
Matrices a diagonal a hlaha ka tlhaho lits'ebetsong tse fapaneng, mohlala:
1. Sekala sa Phetoho ho Litšoantšo tsa Khomphutha
Ho holisa kapa ho fokotsa ntho ka thoko ho li-axes tsa \(x\), \(y\), le \(z\), ho sebelisoa matrix e otlolohileng eo likarolo tsa eona tse otlolohileng li nang le lintlha tsa sekala.
2. Ho fapana ha Lipalopalo
Haeba di-variable tse sa reroang di sa amane, matrix ya covariance e diagonal hobane covariance pakeng tsa di-variable ke lefela.
3. Mohlala o Molele le Boima
Ho ntlafatsa le ho ithuta ka mochini, matrices e otlolohileng hangata e sebelisoa e le matrices ea boima e fanang ka likotlo tse fapaneng ho karolo ka 'ngoe.
Ho koala
Sebopeho sa matrix e otlolohileng ke e 'ngoe ea meaho e bonolo ka ho fetisisa ea matrix empa e le molemo ka ho fetisisa. Matrix ena e khetholloa ka likarolo tsohle tse seng tse otlolohileng tse le zero, ha likarolo tse otlolohileng li ka fapana. Sebopeho sena se etsa hore ts'ebetso ea bohlokoa joalo ka li-determinant, inverse, multiplication le exponentiation e be bonolo haholo. Hase feela hore matrix e otlolohileng ke ea bohlokoa ka khopolo-taba ho algebra e otlolohileng, empa e boetse e sebelisoa haholo lits'ebetsong tse fapaneng tsa lefats'e la nnete, ho tloha lipalo-palo ho ea ho litšoantšo tsa khomphutha.
Ho utloisisa matrices e otlolohileng ke mohato oa pele o matla oa ho ithuta likhopolo tse tsoetseng pele haholoanyane joalo ka li-eigenvalues, li-eigenvectors, le diagonalization, tse leng bohareng ba mekhoa e mengata ea sejoale-joale ea ho bala.
