## Appendix AVectors and Matrices

### A.1 Vectors and Norms

A vector is an array of real or complex numbers. We denote vectors by lowercase bold letters and assume a vector to be a column vector. For example, a vector of length *n* is

$\mathbf{x}=\left[\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill {x}_{2}\hfill \\ \hfill \vdots \hfill \\ \hfill {x}_{n}\hfill \end{array}\right].$

The number of elements in a vector is also known as the dimension of the vector. If the elements *x*_{1}, *x*_{2},... , *x _{n}* are real, then we say that

**x**is a real

*n*-dimensional vector, and we can denote it as $\mathbf{x}\in {\mathbb{R}}^{n}$. On the other hand, if the elements are complex, then we say that

**x**is a complex

*n*-dimensional vector, and it is denoted as $\mathbf{x}\in {\u2102}^{n}$.

Various norms are defined to measure the magnitude of a vector. One such measure is *ℓ _{p}* norm, which is defined as

(A.1.1) $||X|{|}_{p}={[\underset{i=1}{\overset{n}{\mathrm{\Sigma}}}|{x}_{i}{|}^{p}]}^{1/p},$

where *n* is the dimension of ...

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