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行列の分解

コレスキー、LU 分解、QR 分解、特異値分解、行列のジョルダン、フロベニウス、エルミート、およびスミス形式

メモ

MuPAD® Notebook は将来のリリースでは削除される予定です。代わりに MATLAB® ライブ スクリプトを使用してください。

MuPAD Notebook ファイルを MATLAB ライブ スクリプト ファイルに変換するには、convertMuPADNotebook を参照してください。MATLAB ライブ スクリプトは、多少の違いはありますが、MuPAD 機能の大半をサポートします。詳細は、MuPAD Notebook を MATLAB ライブ スクリプトに変換を参照してください。

MuPAD 関数

linalg::factorCholeskyThe Cholesky decomposition of a matrix
linalg::factorLULU-decomposition of a matrix
linalg::factorQRQR-decomposition of a matrix
linalg::frobeniusFormFrobenius form of a matrix
linalg::hermiteFormHermite normal form of a matrix
linalg::inverseLUComputing the inverse of a matrix using LU-decomposition
linalg::jordanFormJordan normal form of a matrix
linalg::smithFormSmith normal form of a matrix
numeric::factorCholeskyCholesky factorization of a matrix
numeric::factorLULU factorization of a matrix
numeric::factorQRQR factorization of a matrix
numeric::singularvaluesNumerical singular values of a matrix
numeric::singularvectorsNumerical singular value decomposition of a matrix
numeric::svdNumerical singular value decomposition of a matrix

例および操作のヒント

Compute Cholesky Factorization

The Cholesky factorization expresses a complex Hermitian (self-adjoint) positive definite matrix as a product of a lower triangular matrix L and its Hermitian transpose LH: A = L LH. The Hermitian transpose of a matrix is the complex conjugate of the transpose of that matrix. For real and symmetric matrices, the transpose coincides with the Hermitian transpose. Thus, the Cholesky factorization of a real symmetric positive definite matrix is A = L LT, where LT is the transpose of L.

Compute LU Factorization

The LU factorization expresses an m×n matrix A as follows: P*A = L*U. Here L is an m×m lower triangular matrix that contains 1s on the main diagonal, U is an m×n matrix upper triangular matrix, and P is a permutation matrix. To compute the LU decomposition of a matrix, use the linalg::factorLU function. For example, compute the LU decomposition of the following square matrix:

Compute QR Factorization

The QR factorization expresses an m×n matrix A as follows: A = Q*R. Here Q is an m×m unitary matrix, and R is an m×n upper triangular matrix. If the components of A are real numbers, Q is an orthogonal matrix. To compute the QR decomposition of a matrix, use the linalg::factorQR function. For example, compute the QR decomposition of the 3×3 Pascal matrix:

Compute Factorizations Numerically

For numeric factorization functions, you can use the HardwareFloats, SoftwareFloats and Symbolic options. For information about these options, see the Compute Determinant Numerically section. For more details, see the help pages of the numeric functions provided for each particular factorization function.

Find Jordan Canonical Form of a Matrix

The Jordan canonical form of a square matrix is a block matrix in which each block is a Jordan block. A Jordan block is a square matrix with an eigenvalue of the original matrix on the main diagonal. A block also can contain 1s on its first superdiagonal. Each Jordan block corresponds to a particular eigenvalue. Single eigenvalues produce 1×1 Jordan blocks. If an n×n square matrix has n linearly independent eigenvectors, the Jordan form of that matrix is a diagonal matrix with the eigenvalues on the main diagonal. For example, create the 3 ×3 Pascal matrix P:

概念

Linear Algebra Library

Use only in the MuPAD Notebook Interface. This functionality does not run in MATLAB.

Numeric Algorithms Library

Use only in the MuPAD Notebook Interface. This functionality does not run in MATLAB.