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The Matrices and Linear Algebra library provides three large sublibraries containing blocks for linear algebra; Linear System Solvers, Matrix Factorizations, and Matrix Inverses. A fourth library, Matrix Operations, provides other essential blocks for working with matrices.
The Linear System Solvers library provides the following blocks for solving the system of linear equations AX = B:
Some of the blocks offer particular strengths for certain classes of problems. For example, the Cholesky Solver block is particularly adapted for a square Hermitian positive definite matrix A, whereas the Backward Substitution block is particularly suited for an upper triangular matrix A.
In the following ex_lusolver_tutex_lusolver_tut model, the LU Solver block solves the equation Ax = b, where
$$A=\left[\begin{array}{ccc}1& -2& 3\\ 4& 0& 6\\ 2& -1& 3\end{array}\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}b=\left[\begin{array}{c}1\\ -2\\ -1\end{array}\right]$$
and finds x to be the vector [-2 0 1]'.
You can verify the solution by using the Matrix Multiply block to perform the multiplication Ax, as shown in the following ex_matrixmultiply_tut1ex_matrixmultiply_tut1 model.
The Matrix Factorizations library provides the following blocks for factoring various kinds of matrices:
Some of the blocks offer particular strengths for certain classes of problems. For example, the Cholesky Factorization block is particularly suited to factoring a Hermitian positive definite matrix into triangular components, whereas the QR Factorization is particularly suited to factoring a rectangular matrix into unitary and upper triangular components.
In the following ex_lufactorization_tutex_lufactorization_tut model, the LU Factorization block factors a matrix A_{p} into upper and lower triangular submatrices U and L, where A_{p} is row equivalent to input matrix A, where
The lower output of the LU Factorization, P, is the permutation index vector, which indicates that the factored matrix A_{p} is generated from A by interchanging the first and second rows.
$${A}_{p}=\left[\begin{array}{ccc}4& 0& 6\\ 1& -2& 3\\ 2& -1& 3\end{array}\right]$$
The upper output of the LU Factorization, LU, is a composite matrix containing the two submatrix factors, U and L, whose product LU is equal to A_{p}.
$$U=\left[\begin{array}{ccc}4& 0& 6\\ 0& -2& 1.5\\ 0& 0& -0.75\end{array}\right]\text{}\text{}\text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}L=\left[\begin{array}{ccc}1& 0& 0\\ 0.25& 1& 0\\ 0.5& 0.5& 1\end{array}\right]$$
You can check that LU = A_{p} with the Matrix Multiply block, as shown in the following ex_matrixmultiply_tut2ex_matrixmultiply_tut2 model.
The Matrix Inverses library provides the following blocks for inverting various kinds of matrices:
In the following ex_luinverse_tutex_luinverse_tut model, the LU Inverse block computes the inverse of input matrix A, where
$$A=\left[\begin{array}{ccc}1& -2& 3\\ 4& 0& 6\\ 2& -1& 3\end{array}\right]$$
and then forms the product A^{-1}A, which yields the identity matrix of order 3, as expected.
As shown above, the computed inverse is
$${A}^{-1}=\left[\begin{array}{ccc}-1& -0.5& 2\\ 0& 0.5& -1\\ 0.6667& 0.5& -1.333\end{array}\right]$$