Vector Norms¶
Our interest is in operators mapping vectors from a model space \(\mathbb{C}^n\) to a data space \(\mathbb{C}^m\).
There are some simple and useful results on relationships between different \(p\)-norms listed in this section. We also discuss some interesting properties of \(l_1\)-norm specifically.
Let \(v \in \mathbb{C}^n\). Let the entries in \(v\) be represented as
where \(r_i = | v_i |\) with the convention that \(\theta_i = 0\) whenever \(r_i = 0\).
The sign vector for \(v\) denoted by \(\sgn(v)\) is defined as
where
For any \(v \in \mathbb{C}^n\) :
Note that whenever \(v_i = 0\), corresponding \(0\) entry in \(\sgn(v)\) has no effect on the sum.
Suppose \(v \in \mathbb{C}^n\). Then
For the lower bound, we go as follows
This gives us
We can write \(l_1\) norm as
By Cauchy-Schwartz inequality we have
Since \(\sgn(v)\) can have at most \(n\) non-zero values, each with magnitude 1,
Thus, we get
Let \(v \in \mathbb{C}^n\). Then
Let \(v \in \mathbb{C}^n\). Let \(1 \leq p, q \leq \infty\). Then
Let \(\OneVec \in \mathbb{C}^n\) be the vector of all ones i.e. \(\OneVec = (1, \dots, 1)\). Let \(v \in \mathbb{C}^n\) be some arbitrary vector. Let \(| v |\) denote the vector of absolute values of entries in \(v\). i.e. \(|v|_i = |v_i| \Forall 1 \leq i \leq n\). Then
Finally since \(\OneVec\) consists only of real entries, hence its transpose and Hermitian transpose are same.
Let \(\OneMat \in \CC^{n \times n}\) be a square matrix of all ones. Let \(v \in \mathbb{C}^n\) be some arbitrary vector. Then
\(k\)-th largest (magnitude) entry in a vector \(x \in \mathbb{C}^n\) denoted by \(x_{(k)}\) obeys
Let \(n_1, n_2, \dots, n_N\) be a permutation of \(\{ 1, 2, \dots, n \}\) such that
Thus, the \(k\)-th largest entry in \(x\) is \(x_{n_k}\). It is clear that
Obviously
Similarly
Thus