In linear algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations.
In general, a system with linear equations and unknowns can be written as
\begin{align} a11x1+a12x2+ … +a1nxn&=b1\\ a21x1+a22x2+ … +a2nxn&=b2\\ & \vdots\\ am1x1+am2x2+ … +amnxn&=bm \end{align}
where
x1,x2,\ldots,xn
a11,a12,\ldots,amn
\begin{bmatrix} a11&a12& … &a1n\\ a21&a22& … &a2n\\ \vdots&\vdots&\ddots&\vdots\\ am1&am2& … &amn\end{bmatrix}
Then the above set of equations can be expressed more succinctly as
Ax=b
where is the coefficient matrix and is the column vector of constant terms.
By the Rouché–Capelli theorem, the system of equations is inconsistent, meaning it has no solutions, if the rank of the augmented matrix (the coefficient matrix augmented with an additional column consisting of the vector) is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has free parameters; hence in such a case there are an infinitude of solutions, which can be found by imposing arbitrary values on of the variables and solving the resulting system for its unique solution; different choices of which variables to fix, and different fixed values of them, give different system solutions.
A first-order matrix difference equation with constant term can be written as
yt+1=Ayt+c,
where is and and are . This system converges to its steady-state level of if and only if the absolute values of all eigenvalues of are less than 1.
A first-order matrix differential equation with constant term can be written as
| dy | |
| dt |
=Ay(t)+c.
This system is stable if and only if all eigenvalues of have negative real parts.