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 \(\mathbf{A} = \mathbf{A}^T\).