Non-Programming Platform for Matrix and Vector Operations in Linear Algebra

A non-programming platform consisting of modules (functions) for matrix and vector operations, and each module carrying out a class of distinct operations in linear algebra, which include (1) matrices and vector operations and manipulations (addition, subtraction, multiplication, inverse, matrix polynomials, merging matrices, transpose, conjugate of complex matrices, rotation matrices, Jacobian matrices, special matrices, and random matrices);

• (2) finding determinants, traces, minors, cofactor matrices, norms, ranks, and condition numbers;
• (3) determining if a matrix is diagonalizable and nilpotent, and determining definiteness (positive definite, positive semidefinite, negative definite, negative semidefinite, indefinite);
• (4) solving linear systems by reducing a matrix to echelon form or reduced row echelon form, determining linear independence and linear combinations, determining a vector in a span, and finding coordinate vectors relative to various bases;
• (6) determining the dimension and a basis for the four fundamental subspaces of a matrix or linear mapping;
• (7) determining a linear mapping, the matrix representation of a linear mapping, and the change of basis matrix from one basis to another;
• (8) computing inner product, distance and angle between two vectors, matrix representation of inner product, determining orthogonal projection in an inner product space, and verifying orthogonal and normal matrices;
• (9) computing eigenvalues and eigenvectors, singular values, characteristic polynomials, and using Gram-Schimdt process to generate an orthogonal or orthonormal set of vectors;
• (10) diagonalizing a matrix or linear operator (real symmetric matrices and quadratic forms) using eigenvalues and eigenvectors;
• (11) matrix decomposition and Jordan canonic forms;
• (12) computing adjoint operators, determining linear forms and dual bases, determining bilinear forms in matrix equations, matrix representation, change of basis matrix for bilinear forms, and diagonalizing symmetric and Hermitian matrices, normal matrices, symmetric bilinear forms, quadratic forms, and Hermitian forms;
• wherein applying each module for its associated operations requires one short line of self-explaining input that consists of necessary elements such as module names (e.g., “ref”, “eig”, “mrg”, “det”), expressions and functions, variables, numbers, keywords, and other related parameters;
• wherein users can access and interact with these modules in many different ways:
• (I) using a typical standalone personal computer (workstation or server) that has these modules installed, and has Windows 10, Unix or Linux, or Mac OS with an Intel or other similar processor of 2.50 GHz frequency (or greater) and 64 bit 4 GB (or greater) RAM;
• (II) through an online web application (already created) with a computer, cell phone, smart phone, tablet or ipad, or other similar devices that have an access to the Internet.