Linear Algebra and Computing

Math Programming


Scientific Computing and Linear Algebra

Table of Contents


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