| Abstract | Faktorizacije matrica od velikog su značaja u teoriji matrica i uopće u numeričkoj linearnoj algebri. Jedan od osnovnih alata u analiziranju i numeričkom rješavanju problema svojstvenih vrijednosti je Schurova dekompozicija. Upravo nam ova dekompozicija daje odgovor na pitanje koliko najviše možemo unitarnom transformacijom sličnosti pojednostaviti matricu\( A ∈ C^{ n×n}\) .
Svojstvene vrijednosti i svojstveni vektori matrice A jednostavno se prenose na njoj
sličnu matricu \(B = S^{-1}AS\), ako je Bx = λx, onda je ASx = λSx. Dakle, za računanje
svojstvenih vrijednosti i svojstvenih vektora matrice A praktično je tražiti transformacije sličnosti koje će dati matricu B koja je jednostavnije strukture.
Unitarna transformacija sličnosti matrice A gornjetrokutastoj matrici dana je Schurovim postupkom dekompozicije matrice. Unitarna transformacija sličnosti S ima prednost zbog nekih teoretskih i numeričkih svojstava, kao što je jednostavnije računanje
inverza matrice transformacije, budući vrijedi \(S^{-1}=S^{∗}\)
te ne povećava normu pertubacije, čuva neke klase matrice. Primjerice, ako je A normalna, hermitska, antihermitska,
unitarna, onda je matrica B \(B = S^{∗}AS)\)
) normalna, hermitska, antihermitska, unitarna, respektivno.
U ovom radu dani su rezultati o Schurovoj dekompoziciji. Postupak Schurove dekompozicije (realne i kompleksne) zasnovan na upotrebi Gram-Schmidtovog postupka
ortogonalizacije ilustriran je kroz primjere, a značaj Schurovog teorema dodatno je istaknut njegovim posljedicama predstavljenim u obliku teorema s kojima smo se već
imali priliku susresti. |
| Abstract (english) | Matrix factorization has a great significance in matrix theory, as well as in Numerical
Linear Algebra. Schur decomposition is considered as one of the basic tools, when it
comes to analyzes and finding a numerical solution to an eigenvalue problem. This decomposition in particular is proven to give an answer to the question, to which level a
matrix \( A ∈ C^{ n×n}\) .
can be simplified by the use of unitary transformations of similarity.
Eigenvalues and eigenvectors of matrix A are transferred to the matrici \(B = S^{-1}AS\),
which is similar to the matrix A. So, if matrices A and B are similar, following equation
Bx = λx implies that ASx = λSx. This process implies that it can be practical to use
a matrix B, which has a simplified structure in order to calculate the eigenvalues and
eigenvectors of matrix A.
The process of unitary transformation of similarity which transforms a matrix A to an
upper triangular matrix is given by the Schur decomposition. Unitary transformation
of similarity S is proven to have some advantages when it comes to its theoretical and
numerical properties. This particular case allows as to calculate the inverse of matrix of
transformation more easily, as \(S^{-1}=S^{∗}\)
, and the norm of perturbations is not enlarged
during the process, as a class of matrix is preserved. For instance, if matrix A is normal, hermitian, antihermitian or unitary, than matrix \(B = S^{∗}AS)\) is also normal,
hermitian, antihermitian or unitary, respectively.
In this paper the results of Schur decomposition are represented. The process of Schur
decomposition (real and complex) is based on the use of the Gram-Schmidt orthogonalization process. Examples represented in this paper have covered both real and complex
cases, so that the process can be fully illustrated. Significance of Schur theorem is stated
in terms of well known theorems which are all results of Schur decomposition process |
