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Real Linear Equations
Real Vectors
Needed by:
Autonomous Continuous-Time Linear Dynamical Systems
Circulant Matrices
Complex Matrices
Confusion Matrices
Covariance Matrix
Data Matrix
Diagonal Constant Matrices
Eigenvalues and Eigenvectors
Matrix Determinant of Product
Matrix Permanents
Matrix Rings
Matrix Similarity
Matrix Trace
Range and Null Space
Real Affine Sets and Linear Equations
Real Matrix Determinants
Real Matrix Exponential
Real Matrix Minors
Real Matrix Nullspace
Real Matrix Polynomials
Real Matrix Range
Real Matrix Rank
Real Similarity Transformations
Real Submatrices
Row Reducer Matrices
Vectors as Matrices
Sheet PDF
Graph PDF

Real Matrices


We compress the notation for linear equations.


A real matrix (matrix, rectangular array) is a two-dimensional array of real numbers. We denote the elements of a matrix in a grid between two rectangular braces, as in

\[ A = \bmat{ A_{11} & A_{12} & \cdots & A_{1n} \\ A_{21} & A_{22} & \cdots & A_{2n} \\ \vdots & & \ddots & \vdots \\ A_{m1} & A_{m2} & \cdots & A_{mn} \\ }. \]

We call $m$ and $n$ the dimensions of the matrix. We call $m$ the height and $n$ the width. If the height of the matrix is the same as the width of the matrix then we call the matrix square. If the height is larger than the width, we call the matrix tall. If the width is larger than the height, we call the matrix wide.

Linear equations

Recall that we are interested in solutions of the linear equations

\[ \begin{aligned} y_1 &= A_{11}x_1 + A_{12}x_2 + \cdots + A_{1n}x_n, \\ y_2 &= A_{21}x_1 + A_{22}x_2 + \cdots + A_{2n}x_n, \\ &\vdots \\ y_n &= A_{m1}x_1 + A_{m2}x_2 + \cdots + A_{mn}x_n. \\ \end{aligned} \]

We have suggestively used the notation $A_{ij}$ for the coefficients of the equations, so they are the entries of $A \in \R ^{m \times n}$.

A primer on matrix-vector products

Using the notation $A \in \R ^{m \times n}$ and $x \in \R ^n$ we want a compressed way to write the above system of linear equations. Define the real matrix-vector product $z \in \R ^m$ of $A$ with $x$ by

\[ z_{i} = \sum_{j = 1}^{n} A_{ij}x_j, \quad i = 1, \dots , m. \]

We denote the matrix vector product $z$ by $Ax$.


We express the above system of linear equations as

\[ y = Ax, \]

where $y = (y_1, \dots , y_n) \in \R ^{n}$ and $x = (x_1, \dots , x_n) \in \R ^n$. The compact notation $y = Ax$ is sometimes called the matrix form of the $m$ linear equations. In this case, $A$ is often called the coefficient matrix.

Note on terminology

The word “matrix” is from the Latin “mater,” meaning mother, and has an old sense in English similar “womb.” The matrix is source of many determinants (discussed elsewhere).

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