The set of real numbers is denoted \(\mathbb{R}\). The set of rational numbers is denoted \(\mathbb{Q}\). The set of integers is denoted \(\mathbb{Z}\).
The set of \(n\)-tuples with real entries is denoted \(\mathbb{R}^n\). Similar definitions hold for \(\mathbb{Q}^n\) and \(\mathbb{Z}^n\).
The set of \(m\times n\) matrices (i.e. matrices with \(m\) rows and \(n\) columns) with real entries is denoted \(\mathbb{R}^{m \times n}\). Similar definitions hold for \(\mathbb{Q}^{m\times n}\) and \(\mathbb{Z}^n\).
All \(n\)-tuples are written as column vectors (i.e. as \(n\times 1\) matrices). An \(n\)-tuple is normally represented by a lowercase Roman letter in boldface. e.g. \(\mathbf{a}\).
Matrices are normally represented by an uppercase Roman letter in boldface. e.g. \(\mathbf{A}\). Column \(j\) of \(\mathbf{A}\) is denoted by \(\mathbf{A}_j\).
Scalars are usually represented by lowercase Greek letters. e.g. \(\lambda\), \(\beta\). In contexts where there is no confusion, a \(1\times 1\) matrix can be treated as a scalar.
A matrix or tuple consisting of all zeros is simply denoted \(\mathbf{0}\) with the dimension inferred from the context.
For a matrix \(\mathbf{A}\), \(\mathbf{A}^\mathsf{T}\) denotes the transpose of \(\mathbf{A}\). For an \(n\)-tuple \(\mathbf{x}\), \(\mathbf{x}^\mathsf{T}\) denotes the transpose of \(\mathbf{x}\).
For an \(n\)-tuple \(\mathbf{u}\), \(u_i\) denotes the \(i\)th entry (or component) of \(\mathbf{u}\) for \(i = 1,\ldots, n\). Note that boldface is not used in \(u_i\).
When \(\mathbf{u}\) and \(\mathbf{v}\) are \(n\)-tuples, \(\mathbf{u} \geq \mathbf{v}\) (or \(\mathbf{u}^\mathsf{T} \geq \mathbf{v}^\mathsf{T}\)) means \(u_i \geq v_i\) for \(i = 1,\ldots, n\). Similar definitions hold for \(\mathbf{u} \leq \mathbf{v}\) and \(\mathbf{u} = \mathbf{v}\).
Superscripts are used for indexing tuples or vectors. For example, if \(\mathbf{u}^1,\mathbf{u}^2 \in \mathbb{R}^3\), then \(\mathbf{u}^1\) and \(\mathbf{u}^2\) are elements of \(\mathbb{R}^3\).