Basic operations and properties of matrices

One of the research aims of numerical linear algebra is to look for an approximate solution to mathematical problems in a continuous version. These mathematical problems arise in engineering and natural science. Numerical linear algebra as a fundamental computational science tool is frequently used in image and signal processing, data mining, computational finance, bioinformatics, telecommunication, fluid dynamics, and material science simulation. Matrices, one of the fundamental concepts in numerical linear algebra, are ubiquitous in natural science and engineering fields. In this paper, we first take an investigation of essential concepts for matrices. The definition of the production of a matrix and a vector and the definition of the production of two matrices are introduced. The fundamental invariant properties such as range, null space, and rank of a matrix are discussed. We also present several results related to orthogonal properties. Several results related to different norms on matrices are presented. At last, an impact on Householder transformation is presented.