Preface
This is a tutorial made solely for the purpose of education and it was designed for students taking Applied Math 0340. It is primarily for students who have some experience using Mathematica. If you have never used Mathematica before and would like to learn more of the basics for this computer algebra system, it is strongly recommended looking at the APMA 0330 tutorial. As a friendly reminder, don'r forget to clear variables in use and/or the kernel.
Finally, the commands in this tutorial are all written in bold black font, while Mathematica output is in regular fonts. This means that you can copy and paste all comamnds into Mathematica, change the parameters and run them. You, as the user, are free to use the scripts to your needs for learning how to use the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.
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Square matrices
A very important class of matrices constitute so called square matrices that have the same number of rows as columns. In computer graphics, square matrices are used for transformations. An n-by-n matrix is known as a square matrix of order n. Square matrices of the same order can be not only added, subtructed or multiplied by a constant, but also multiplied.
Theorem: If A is an \( n \times n \) matrix, then the following statements are equivalent.
- A is invertible.
- \( {\bf A}\,{\bf x} = {\bf 0} \) has only the trivial solution.
- The reduced row echelon form (upon application of Gauss--Jordan elimination) is the identity matrix.
- A has nullity/kernel 0. ■
- A has rank n.
- A is expressible as a product of elementary matrices.
- \( {\bf A}\,{\bf x} = {\bf b} \) is consistent for every \( n \times 1 \) matrix b.
- \( {\bf A}\,{\bf x} = {\bf b} \) has exactly one solution for every \( n \times 1 \) matrix b.
- \( \det ({\bf A} ) \ne 0 .\)
- \( \lambda =0 \) is not an eigenvalue of A.
- The row vectors of A span \( \mathbb{R}^n . \)
- The column vectors of A form a basis for \( \mathbb{R}^n . \)
How to define Vectors
How to define Matrices
Basic Operations with Matrices
Linear Systems of Equations
Determinants and Inverse Matrices
Special Matrices
Eigenvalues and Eigenvectors
Generalized Eigenvectors
Diagonalization Procedure
Sylvester Formula
The Resolvent Method
Spectral Decomposition Method
Positive Matrices
Roots
Miscellaneous
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