3-Dimensional Matrix Mathematical Operations

Abstract

This thesis is an attempt to develop mathematically consistent procedures to perform Three-Dimensional (3-D) matrix math. 3-D matrix operations for Right-multiplication, Left-multiplication, Left-Identity, Right-Identity, Left-Inverse, Right-Inverse, and Eigenvalue/Eigenvector matrix are explained.The innovative technique used here is to separate Three-Dimensional (3-D) matrix arrays into multiple Two-dimensional (2-D) matrix arrays to simplify performing 3-D array mathematics. A n x m x A: 3-D matrix is separated into k, n x m 2-D matrices. Then 2-D array math techniques are used to produce a 3-D Matrix result. The purpose of the present study is to adjust the accepted procedures used to perform 2-D matrix operations for application to 3-D matrices. These new procedures will be explained with a select group of examples. In this project techniques to successfully perform the basic 3-D Matrix manipulations that are derived from the rules for 2-D matrix operation are explored. The process of performing these 3-D matrix operations follow from the definition for Multiplication of a 3-D matrix box. Multiplication is performed by taking Top face to Bottom face matrices of the first box and multiply with the Front face to Back face matrices of the second box which equals the Top face to Bottom face of the result matrix. And similarly define operation to calculate identity, inverse, and eigenvector matrix. Followed by numerical example for each of these operations.