## Linear Algebra and Least Squares

### Linear Algebra Blocks

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.

### Linear System Solvers

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 adapted for a square Hermitian positive definite matrix A, whereas the Backward Substitution block is suited for an upper triangular matrix A.

#### Solve AX=B Using the LU Solver Block

In the following ex_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_tut1 model. ### Matrix Factorizations

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 suited to factoring a Hermitian positive definite matrix into triangular components, whereas the QR Factorization is suited to factoring a rectangular matrix into unitary and upper triangular components.

#### Factor a Matrix into Upper and Lower Submatrices Using the LU Factorization Block

In the following ex_lufactorization_tut model, the LU Factorization block factors a matrix Ap into upper and lower triangular submatrices U and L, where Ap 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 Ap 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 Ap.

`$U=\left[\begin{array}{ccc}4& 0& 6\\ 0& -2& 1.5\\ 0& 0& -0.75\end{array}\right]\text{​}\text{​}\text{​}\text{​}\text{​}\text{​}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}L=\left[\begin{array}{ccc}1& 0& 0\\ 0.25& 1& 0\\ 0.5& 0.5& 1\end{array}\right]$`

You can check that LU = Ap with the Matrix Multiply block, as shown in the following ex_matrixmultiply_tut2 model. ### Matrix Inverses

The Matrix Inverses library provides the following blocks for inverting various kinds of matrices:

#### Find the Inverse of a Matrix Using the LU Inverse Block

In the following ex_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-1A, which yields the identity matrix of order 3, as expected. As shown above, the computed inverse is

`${A}^{-1}=\left[\begin{array}{ccc}-1& -0.5& 2\\ 0& 0.5& -1\\ 0.6667& 0.5& -1.333\end{array}\right]$`