Matrix
Functions & Advanced AlgebraA matrix is a rectangular array of numbers arranged in rows and columns, used to organize data and perform transformations.
Definition
A matrix is a grid of numbers organized in rows (horizontal) and columns (vertical). It is like a spreadsheet of numbers packed into a mathematical object.
Example
A $2\times 3$ matrix: $\begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\end{bmatrix}$. It has $2$ rows and $3$ columns. The entry in row $1$, column $2$ is $2$. Matrices are labeled with capital letters like $A$, $B$, $C$.
Key Insight
Matrices organize information compactly. A system of equations, a network of roads, or a color image can all be represented as matrices, making them one of the most versatile tools in all of mathematics.
Definition
A matrix is a rectangular array with $m$ rows and $n$ columns, called an $m \times n$ matrix. The entry in row $i$, column $j$ is denoted $a_{ij}$. Matrices can be added (if same dimensions), multiplied by scalars, and multiplied together (if dimensions are compatible).
Example
Matrix $A = \begin{bmatrix}2 & -1\\0 & 3\end{bmatrix}$ is $2\times 2$. Entry $a_{12} = -1$. Matrix $B = \begin{bmatrix}1\\4\end{bmatrix}$ is $2\times 1$ (a column vector). The product $AB$ is defined ($2\times 2$ times $2\times 1$ gives $2\times 1$).
Key Insight
Matrices represent linear transformations. The product $AB$ represents the composition of the transformation $B$ first, then $A$. This is why matrix multiplication is not commutative: order of transformations matters.
Definition
An $m \times n$ matrix over a field $F$ is an element of the vector space $M_{m,n}(F)$, of dimension $mn$. The set $M_n(F)$ of square matrices forms a ring (non-commutative for $n \ge 2$). Matrices represent linear maps between finite-dimensional vector spaces, and every linear map $T: F^n \to F^m$ corresponds to an $m \times n$ matrix relative to chosen bases.
Example
The matrix representation of a linear map depends on the choice of basis. Changing basis by an invertible matrix $P$ transforms $A \to P^{-1}AP$ (similarity transformation). Similar matrices represent the same linear map in different bases.
Key Insight
Jordan normal form shows that every complex square matrix is similar to a canonical "almost diagonal" form. The Jordan blocks encode the geometric and algebraic multiplicities of eigenvalues, revealing the structure of every linear transformation over $\mathbb{C}$.