## Linear Algebra and Computing

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Scientific Computing and Linear Algebra

### Introduction

This post just remember several linear algebra properties that are relevant for scientific programming.

### Reducible Matrix

Definition: An square matrix A is reducible if there is a matrix P (basis change) such that, $PAP^{T}=\begin{bmatrix} B_{11} & B_{11}\\ 0 & B_{22} \end{bmatrix}$

This property is very useful because the typical linear system solution $$Ay=c$$ is now more easy to resolve in terms of computing because $\begin{bmatrix} B_{11} & B_{11}\\ 0 & B_{22} \end{bmatrix}\begin{bmatrix} y_{1} \\ y_{2} \end{bmatrix}=\begin{bmatrix} c_{1}\\ c_{2} \end{bmatrix}$

The subsystem $$B_{22}y_{2}=c_{2}$$ is indepentent of the other B matrices with the logical computational improvement.

### Transpose Matrix

The transpose matrix $$\mathbf{B}$$ of a given matrix $$\mathbf{A}$$ is a matrix with elements: $[b_{ij}] = [a_{ji}] ,\quad\quad \forall i =1, \ldots , m ,\quad\quad \forall j = 1, \ldots , n$

The transpose matrix of $$\mathbf{A}$$ is usually noted by $$\mathbf{A}^T$$.

### Ortogonal Matrix

A square matrix $$\mathbf{A}$$ is orthogonal when the matrix product with the transpose matrix has the following property: $\mathbf{A} \mathbf{A^T} = \mathbf{A^T} \mathbf{A} = \mathbf{I}$

### Symmetric Matrix

A square matrix $\mathbf{A}$ is symmetric when $$[a_{ij}] = [a_{ji}]$$. Therefore, a matrix is symmetric if $$\mathbf{A} = \mathbf{A}^T$$.