May 14, 2021 Julia
Matrix decomposition is the product of breaking down a matrix into several matrices and is a core concept in linear algege for.
The following table summarizes several matrix decomposition methods implemented in Julia. Specific functions can refer to the Linear Algebra chapter of the standard library documentation.
| Cholesky | Cholesky decomposes |
| CholeskyPivoted | The primary Cholesky decomposes |
| LU | LU decomposition |
| LUTridiagonal | The LU factor decomposition of the triangular matrix |
| UmfpackLU | LU decomposition of sparse matrices (calculated using UMFPACK) |
| Qr | QR decomposition |
| QRCompactWY | The compact WY form of QR decomposition |
| QRPivoted | The primary QR is decomposed |
| Hessenberg | Hessenberg decomposes |
| Eigen | Feature decomposition |
| Svd | Singular value decomposition |
| GeneralizedSVD | Broad singular value decomposition |
Special matrices with symmetrical structures are often encountered in linear algegexuals, which are often associated with matrix decomposition. Julia has a very rich range of special matrix types built in to quickly perform specific operations on special matrices.
The following table summarizes the special matrix types in Julia, which also contain some of the operations that have been optimized in LAPACK.
| Hermitian | Hermit matrix |
| Triangular | Top/Bottom Triangle Matrix |
| Tridiagonal | Triangular matrix |
| SymTridiagonal | Symmetrical triangular moment |
| Bidiagonal | Up/down double verse matrix |
| Diagonal | The verse matrix |
| UniformScaling | Scale the matrix |
| Matrix type | + | - | * | \ | Other optimized functions |
|---|---|---|---|---|---|
| Hermitian | Xy | inv, sqrtm, expm | |||
| Triangular | Xy | Xy | inv, det | ||
| SymTridiagonal | Ⅹ | Ⅹ | XZ | Xy | eigmax/min |
| Tridiagonal | Ⅹ | Ⅹ | XZ | Xy | |
| Bidiagonal | Ⅹ | Ⅹ | XZ | Xy | |
| Diagnoal | Ⅹ | Ⅹ | Xy | Xy | inv, det, logdet, / |
| UniformScaling | Ⅹ | Ⅹ | Xyz | Xyz | / |
Legend:
| Ⅹ | The matrix-matrix operation has been optimized |
|---|---|
| Y | The matrix-vector operation has been optimized |
| Z | The matrix-standard operation has been optimized |
| Matrix type | LAPACK | eig | eigvals | eigvecs | Svd | svdvals |
|---|---|---|---|---|---|---|
| Hermitian | HE | Abc | ||||
| Triangular | Tr | |||||
| SymTridiagonal | St | A | Abc | AD | ||
| Tridiagonal | GT | |||||
| Bidiagonal | BD | A | A | |||
| Diagonal | DI | A |
Legend:
| A | Optimized for finding feature values and/or feature vectors | eigvals (M) for example |
|---|---|---|
| B | The look for ilth to ihth feature values has been optimized | eigvals(M, il, ih) |
| C | The feature values that are found between the .vl, vh are optimized | eigvals(M, vl, vh) |
| D | The feature vector corresponding to the look-for feature value x ,... x1, x2, x2, has been optimized | eigvecs(M, x) |
A
UniformScaling
represents the scalarian number of scalables for a unit
λ*I
. U
nit unit
I
is defined as a constant and is
UniformScaling
T
he dimensions of these operators are
-
of
*
general
\
size and can match the
+
in other binary operators, such as . T
his
A+I
be a square array for
A
- I and A
A-I
Multiplication
I
use unit operator I are an empty operation (unless the scaling factor is one), so there is little overhead.