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Linear Algebra

This is a summary of the linear algebra concepts used in the course:

Matrices and Vectors

  • $A \in \mathbb{R}^{m \times n}$: matrix with $m$ rows and $n$ columns:

$$ A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix} $$

  • $x \in \mathbb{R}^{n}$: (column) vector with $n$ rows and $1$ column:

$$ x = \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{bmatrix} $$

  • A row vector is a matrix with $1$ row and $n$ columns;
  • A scalar is a matrix with $1$ row and $1$ column (i.e. a number).

Matrix Transpose and Matrix Product

  • Transpose of a matrix $A \in \mathbb{R}^{m \times n}$ is the matrix $A^T \in \mathbb{R}^{n \times m}$ such that $A^T_{ij} = A_{ji}$;
  • Matrix $A$ is symmetric if $A = A^T$;
  • The product of two matrices $A \in \mathbb{R}^{m \times n}$ and $B \in \mathbb{R}^{n \times p}$ is the matrix $C \in \mathbb{R}^{m \times p}$:

$$ C = AB \quad \text{where} \quad C_{ij} = \sum_{k=1}^n A_{ik} B_{kj} $$

  • The inner product (or dot product) of two vectors $x, y \in \mathbb{R}^n$ is the scalar $x^T y \in \mathbb{R}$:

$$ <x, y> = x^T y = y^T x = \sum_{i=1}^n x_i y_i $$

  • The outer product of two vectors $x, y \in \mathbb{R}^n$ is the matrix $xy^T \in \mathbb{R}^{n \times n}$:

$$ xy^T = \begin{bmatrix} x_1 y_1 & x_1 y_2 & \dots & x_1 y_n \\ x_2 y_1 & x_2 y_2 & \dots & x_2 y_n \\ \vdots & \vdots & \ddots & \vdots \\ x_n y_1 & x_n y_2 & \dots & x_n y_n \end{bmatrix} $$

  • The Hadamard/Schur product of two vectors $x, y \in \mathbb{R}^n$ is the vector $x \circ y \in \mathbb{R}^n$ such that $(x \circ y)_i = x_i y_i$:

$$ x \circ y = \begin{bmatrix} x_1 y_1 \\ x_2 y_2 \\ \vdots \\ x_n y_n \end{bmatrix} $$

Properties

  • Associativity: $(AB)C = A(BC)$;
  • It is not commutative: $AB \neq BA$ - unless $A$ or $B$ is a scalar;
  • Transpose of a product: $(AB)^T = B^T A^T$;
  • Transpose of sum: $(A + B)^T = A^T + B^T$.

Norms

  • The norm of a vector is its length or magnitude;
  • The euclidean norm (or $l_2$ norm) of a vector $x \in \mathbb{R}^n$ is:

$$ |x|2 = \sqrt{\sum{i=1}^n x_i^2} = \sqrt{x^T x} = \sqrt{<x, x>} $$

  • More generally, the $l_p$ norm of a vector $x \in \mathbb{R}^n$ is:

$$ |x|p = \left(\sum{i=1}^n |x_i|^p\right)^{1/p} $$

  • The $l_1$ norm of a vector $x \in \mathbb{R}^n$ is: $|x|1 = \sum{i=1}^n |x_i|$;
  • The $l_\infty$ norm of a vector $x \in \mathbb{R}^n$ is: $|x|_\infty = \max_i |x_i|$.

Special Kinds of Matrices and Vectors

  • Diagonal matrix: a matrix $D \in \mathbb{R}^{n \times n}$ such that $D_{ij} = 0$ for $i \neq j$;
  • Identity matrix: a diagonal matrix $I \in \mathbb{R}^{n \times n}$ such that $I_{ii} = 1$ for all $i$;
    • $AI = IA = A$ for any matrix $A \in \mathbb{R}^{m \times n}$ - neutral element of matrix multiplication;
  • Upper triangular matrix: a matrix $U \in \mathbb{R}^{n \times n}$ such that $U_{ij} = 0$ for $i &gt; j$;
  • Lower triangular matrix: a matrix $L \in \mathbb{R}^{n \times n}$ such that $L_{ij} = 0$ for $i &lt; j$.

Eigendecomposition

  • Eigenvector of a matrix $A \in \mathbb{R}^{n \times n}$ is a nonzero vector $\vec{v} \in \mathbb{R}^n$ such that $A \vec{v} = \lambda \vec{v}$ for some scalar $\lambda \in \mathbb{R}$;

$$ Ax = \lambda \vec{v} \quad \Leftrightarrow \quad (A - \lambda I) \vec{v} = 0 $$

  • The eigenvalues of a diagonal matrix are its diagonal elements;
  • The matrix trace is the sum of its diagonal elements: $\text{tr}(A) = \sum_{i=1}^n A_{ii} = \sum_{i=1}^n \lambda_i$;
  • The matrix determinant is the product of its eigenvalues: $\text{det}(A) = |A| = \prod_{i=1}^n \lambda_i$;
    • $|A| = |A^T|$;
    • $|AB| = |A| |B|$;
    • $|\alpha A| = \alpha^n |A|$.

Matrix Inverse

  • A matrix is invertible if there exists a matrix $B$ such that $AB = BA = I$;
  • The inverse of a matrix $A \in \mathbb{R}^{n \times n}$ is the matrix $A^{-1} \in \mathbb{R}^{n \times n}$ such that $AA^{-1} = A^{-1}A = I$;
  • $det(A) \neq 0 \quad \Leftrightarrow \quad A \text{ is invertible}$;
  • $det(A^{-1}) = \frac{1}{det(A)}$;
  • If $A$ is invertible, then $Ax = b$ has a unique solution $x = A^{-1} b$ for any $b$;
  • Computational cost of inverting a matrix is $O(n^3)$.

Properties:

  • $(A^{-1})^{-1} = A$;
  • $(AB)^{-1} = B^{-1} A^{-1}$;
  • $(A^T)^{-1} = (A^{-1})^T$.

Quadratic Form and Positive Semidefinite Matrices

Given a matrix $A \in \mathbb{R}^{n \times n}$ and a vector $x \in \mathbb{R}^n$, the quadratic form is:

$$ x^T A x = \sum_{i=1}^n \sum_{j=1}^n A_{ij} x_i x_j \quad \text{where} \quad A \in \mathbb{R}^{n \times n} $$

This can be written as:

$$ \begin{bmatrix} x_1 & x_2 & \dots & x_n \end{bmatrix} \begin{bmatrix} A_{11} & A_{12} & \dots & A_{1n} \\ A_{21} & A_{22} & \dots & A_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ A_{n1} & A_{n2} & \dots & A_{nn} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} $$

  • A positive semidefinite matrix is a symmetric matrix $A \in \mathbb{R}^{n \times n}$ such that $x^T A x \geq 0$ for all $x \in \mathbb{R}^n$;
    • $A$ is positive semidefinite $\quad \Leftrightarrow \quad \lambda_i(A) \geq 0$ for all $i$;
  • A positive definite matrix is a symmetric matrix $A \in \mathbb{R}^{n \times n}$ such that $x^T A x &gt; 0$ for all $x \in \mathbb{R}^n$;
    • $A$ is positive definite $\quad \Leftrightarrow \quad \lambda_i(A) &gt; 0$ for all $i$.