# $p$-th matricial numerical range

## Definition

The $p$-th matricial numerical range is the special case of $(p,k)$-numerical range for $k=1$.

Let $M_n$ will be the set of all matrices of dimension $n$ and by $\mathrm{U}_n$ it will be denoted the set of all unitary matrices of dimension $n$. The $p$-th numerical range (also known as matricial numerical range ) of the matrix $A \in M_n$ is defined as 

$W(p:A) \coloneqq \left\{ X^\dagger A X: C \in \mathbb{C}^{n \times p}, X^\dagger X = \1_p \right\}.$

We can observe that $W(n:A) = \{ U^A U: X _n \}$ and $W(1:A) = W(A)$ as standard numerical range. In particular, many generalizations of standard numerical range $W(A)$ are actually the ranges of certain scalar-valued functions on $W(p:A)$ (see , ) so it is worthwhile to study the properties of $W(p:A)$.

## Convexity of $W(p:A)$

Let $A \in M_n$ will be any matrix of dimension $n$, then in general case the set $W(p:A)$ is non-convex . The following theorems (see ) give us the conditions to matrix $A$ and its eigenvalues so as to the set $W(p:A) will be convex. ### Theorem Let$A \in M_n$will be Hermitian matrix with eigenvalues$\lambda_1 \ge \ldots \ge \lambda_n$. The set$W(p:A)$is convex if and only if$\lambda_1 = \lambda_p$and$\lambda_{n-p+1} = \lambda_n$. We can see that if$A \in M_n$is Hermitian and$p>n/2$, when$W(p:A)$is convex if and only if$A$is a scalar matrix. We can see other theorems involving convexity of$W(p:A)$in . ## Properties Let$1<p<n$. - If all$X \in W(p:A)$are scalar matrices, then$A$is a scalar matrix. Opposite implication is obvious; - All$X \in W(p:A)$are Hermitian if and only if$A$is Hermitian; - All$X \in W(p:A)$are normal if and only if$A\$ is essentially Hermitian.

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