1. What is a matrix
A matrix is a rectangular array of numbers arranged in rows and columns. We write an $m imes n$ matrix $A$ as:
$$A = egin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ dots & dots & \ddots & dots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$
where $a_{ij}$ is the entry in row $i$, column $j$. Matrices encode linear transformations, systems of equations, data tables, graphs, and more.
Core vocabulary
- Dimension: $m$ rows by $n$ columns.
- Square matrix: $m = n$.
- Vector: an $m imes 1$ column or $1 imes n$ row matrix.
- Equality: $A = B$ iff same size and $a_{ij}=b_{ij}$ for all $i,j$.
2. Important types
| Type | Definition | Notation |
|---|---|---|
| Zero | All entries 0 | $0_{m imes n}$ |
| Identity | 1s on diagonal, 0s elsewhere | $I_n$ |
| Diagonal | Nonzero only on $a_{ii}$ | $\operatorname{diag}(d_1,\dots,d_n)$ |
| Upper triangular | $a_{ij}=0$ for $i>j$ | |
| Lower triangular | $a_{ij}=0$ for $i | |
| Symmetric | $A^T = A$ | |
| Skew-symmetric | $A^T = -A$ | |
| Orthogonal | $Q^T Q = I$ | columns are orthonormal |
| Idempotent | $A^2 = A$ | projectors |
| Invertible | Exists $A^{-1}$ with $AA^{-1}=I$ | $\det A eq 0$ |
3. Operations and properties
Addition and scalar multiplication
$(A+B)_{ij}=a_{ij}+b_{ij}$, $(cA)_{ij}=c a_{ij}$. Same size required for addition.
Properties: commutative $A+B=B+A$, associative, distributive $c(A+B)=cA+cB$.
Matrix multiplication
For $A$ ($m imes k$) and $B$ ($k imes n$): $(AB)_{ij} = \sum_{t=1}^k a_{it} b_{tj}$.
Not commutative in general: $AB eq BA$. Associative: $(AB)C = A(BC)$. Distributive over addition.
Transpose
$(A^T)_{ij}=a_{ji}$. Properties: $(AB)^T = B^T A^T$, $(A^T)^T = A$.
Trace
$\operatorname{tr}(A)=\sum_i a_{ii}$. $\operatorname{tr}(AB)=\operatorname{tr}(BA)$.
Interactive: addition
Interactive: multiplication (2x2)
4. Determinant, inverse, rank
Determinant measures volume scaling. For 2x2: $\detegin{bmatrix}a&b\c&d\end{bmatrix}=ad-bc$.
For 3x3 use rule of Sarrus or cofactor expansion.
Inverse: $A^{-1}$ exists iff $\det A eq 0$. For 2x2: $A^{-1} = rac1{ad-bc}egin{bmatrix}d&-b\-c&a\end{bmatrix}$.
Rank: dimension of column space. $\operatorname{rank}(A) \le \min(m,n)$. Full rank means invertible for square matrices.
Interactive: determinant and inverse (2x2)
5. Eigenvalues and eigenvectors
$Av = \lambda v$ for nonzero $v$. $\lambda$ is eigenvalue, $v$ eigenvector.
Characteristic polynomial: $p(\lambda)=\det(A-\lambda I)=0$.
Diagonalization: if $A$ has $n$ independent eigenvectors, $A = PDP^{-1}$ with $D=\operatorname{diag}(\lambda_i)$. Symmetric real matrices are orthogonally diagonalizable.
6. Decompositions
- LU: $PA = LU$, with $L$ lower triangular, $U$ upper. Used for solving systems efficiently.
- QR: $A = QR$, $Q$ orthogonal, $R$ upper triangular. Stable for least squares.
- SVD: $A = U \Sigma V^T$. Works for any $m imes n$. $\Sigma$ contains singular values. Foundation of PCA, compression, pseudoinverse.
- Eigendecomposition: $A = PDP^{-1}$ when diagonalizable.
7. Solving linear systems
System $Ax=b$. If $A$ invertible: $x = A^{-1}b$. In practice use Gaussian elimination or $x = A ackslash b$ in Octave.
Consistency: solution exists iff $\operatorname{rank}(A)=\operatorname{rank}([A|b])$. Unique iff rank equals $n$.
8. GNU Octave lab
Octave syntax mirrors MATLAB. Copy these into Octave.
% Basics A = [1 2; 3 4] B = [5 6; 7 8] A + B 2 * A A * B A' % transpose % Determinant, inverse, rank, trace det(A) inv(A) rank(A) trace(A) % Solve Ax = b b = [5; 11] x = A \ b % preferred over inv(A)*b % Eigen [V, D] = eig(A) % V columns are eigenvectors, D diagonal eigenvalues % LU decomposition with partial pivoting [L, U, P] = lu(A) P*A - L*U % should be near zero % QR [Q, R] = qr(A) Q'*Q % identity check % SVD [U, S, V] = svd(A) U*S*V' - A % reconstruction error % Create special matrices I = eye(3) Z = zeros(2,4) O = ones(3) D = diag([2, -1, 4]) R = rand(3) % uniform random % Symmetric positive definite test M = A'*A eig(M) % all >0
Octave tips
- Use semicolon to suppress output.
- Indexing starts at 1:
A(2,1)is row 2 col 1. - Colon ranges:
A(1:2, :)first two rows. - Elementwise ops use dot:
A .* B,A .^ 2.
9. Why matrices matter
- Computer graphics: rotations, scaling, projections via $3 imes3$ and $4 imes4$ matrices.
- Machine learning: data matrices $X$, weights $W$, predictions $XW$.
- Networks: adjacency matrices encode graphs.
- Differential equations: state transition $x' = Ax$.
- Statistics: covariance matrices, PCA via SVD.
10. Practice problems
- Compute $AB$ for $A=egin{bmatrix}1&2&3\0&1&4\end{bmatrix}$, $B=egin{bmatrix}1&0\-1&2\3&1\end{bmatrix}$.
- Find $\det$ and $A^{-1}$ for $A=egin{bmatrix}2&5\1&3\end{bmatrix}$.
- Show that if $Q$ is orthogonal, $\|Qx\|=\|x\|$.
- In Octave, generate random $4 imes4$ $A$, compute its SVD, verify reconstruction error norm is $<10^{-12}$.
- Diagonalize $A=egin{bmatrix}4&1\2&3\end{bmatrix}$ using
eig.