Questions tagged [matrix-equations]

Definition: Let $~v_1,~v_2,~\cdots~,v_n~ $ and $~b~$ be vectors in $~\mathbb{R^n}~$. Consider the vector equation $$x_1~v_1+x_2~v_2+~\cdots~+x_n~v_n=b~$$This is equivalent to the matrix equation$$~Ax=b~$$

where $~~A=\begin{pmatrix} \cdot & \cdot & \cdots & \cdot \\ v_1 & v_2 & \cdots & v_n \\ \cdot & \cdot & \cdots & \cdot \\ \end{pmatrix};~~ x=\begin{pmatrix} x_1 \\ x_2\\ \cdots\\ x_n \end{pmatrix} ~~\text{and}~~ b=\begin{pmatrix} b_1 \\ b_2\\ \cdots\\ b_n \end{pmatrix}$

Since a matrix equation $ ~AX=B~$ (where $ ~X~$ is a column vector of variables) is equivalent to a system of linear equations, we can use the same methods we have used on systems of linear equations to solve matrix equations. Namely:

$(1.)~~$ Write down the augmented matrix $ ~A \vdots B$.

$(2.)~~$ Row-reduce to a new augmented matrix $~ \overline A \vdots \overline B~$ in row echelon form.

$(3.)~~$ Use this new matrix to write a matrix equation equivalent to the original one.

$(4.)~~$ Use this new, equivalent matrix equation to find the solutions to the original equation.

In mathematics, matrix equation (which is a system of linear equations) is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system.

Very often, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring. Integer linear programming is a collection of methods for finding the "best" integer solution (when there are many). Gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure.