Module I·Article II·~2 min read
Matrices: Operations, Types, Rank
Complex Numbers, Matrices, and Determinants
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Matrix as a Tool
A matrix is a rectangular array of numbers. Notation: $A = (a_{ij})$, $i = 1,\ldots,m$ (rows), $j = 1,\ldots,n$ (columns). $A$ is a matrix of size $m \times n$.
Matrices arose as a convenient way to record systems of linear equations. Today they are the primary object of linear algebra and computational mathematics.
Matrix Operations
Addition: $(A+B){ij} = a{ij} + b_{ij}$. Requires the same size.
Multiplication by a scalar: $(\alpha A){ij} = \alpha \cdot a{ij}$.
Matrix multiplication: $C = AB$, where $c_{ij} = \sum_k a_{ik} \cdot b_{kj}$. Requirement: $A$ of size $m \times k$, $B$ of size $k \times n$. Result $C$ of size $m \times n$.
Multiplication is not commutative: $AB \ne BA$ in general! This is a fundamental difference between matrices and numbers.
Transposition: $(A^T){ij} = a{ji}$. Properties: $(AB)^T = B^T A^T$.
Special Types of Matrices
Identity matrix $E$ (or $I$): $e_{ij} = 1$ when $i = j$, $0$ when $i \ne j$. $AE = EA = A$.
Diagonal: $a_{ij}=0$ when $i \ne j$.
Triangular: lower ($a_{ij} = 0$ when $i < j$) or upper ($a_{ij} = 0$ when $i > j$). Determinant = product of diagonal elements.
Symmetric: $A = A^T$ ($a_{ij} = a_{ji}$). Covariance matrices are symmetric.
Orthogonal: $A^T A = E$. Columns form an orthonormal basis. $\det A = \pm 1$. Transformations are rotations and reflections.
Rank of a Matrix
The rank of $A$ is the maximum order of a nonzero minor, which is also the dimension of the row (and column) space. $\operatorname{rank}(A) = \operatorname{rank}(A^T)$.
Elementary row operations (permutation, multiplication by a scalar, adding a multiple of another row) do not change the rank.
The Gauss method (row reduction to row-echelon form) is the standard algorithm for finding the rank.
Theorem: $\operatorname{rank}(AB) \leq \min(\operatorname{rank} A, \operatorname{rank} B)$.
Inverse Matrix
$A^{-1}$ exists ($A$ is invertible or nonsingular) if and only if $\det A \ne 0$, i.e., $\operatorname{rank} A = n$.
$A A^{-1} = A^{-1} A = E$. $(AB)^{-1} = B^{-1} A^{-1}$.
Computation: $[A|E] \rightarrow [E|A^{-1}]$ using the Gauss–Jordan method.
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