Let $\Theta =(\theta _{j,k})_{3\times 3}$ be a nondegenerate real skew-symmetric $3\times 3$ matrix, where $\theta _{j,k}\in [0,1).$ For any $\varepsilon>0$ , we prove that there exists $\delta>0$ satisfying the following: if $v_1,v_2,v_3$ are three unitaries in any unital simple separable $C^*$ -algebra A with tracial rank at most one, such that

$$\begin{align*}\|v_kv_j-e^{2\pi i \theta_{j,k}}v_jv_k\|<\delta \,\,\,\, \mbox{and}\,\,\,\, \frac{1}{2\pi i}\tau(\log_{\theta}(v_kv_jv_k^*v_j^*))=\theta_{j,k}\end{align*}$$

$\tau \in T(A)$

$j,k=1,2,3,$

$\log _{\theta }$

$\theta \in [0, 1)$

$\tilde {v}_1,\tilde {v}_2,\tilde {v}_3\in A$

$$\begin{align*}\tilde{v}_k\tilde{v}_j=e^{2\pi i\theta_{j,k} }\tilde{v}_j\tilde{v}_k\,\,\,\,\mbox{and}\,\,\,\,\|\tilde{v}_j-v_j\|<\varepsilon,\,\,j,k=1,2,3.\end{align*}$$

The same conclusion holds if $\Theta $ is rational or nondegenerate and A is a nuclear purely infinite simple $C^*$ -algebra (where the trace condition is vacuous).

If $\Theta $ is degenerate and A has tracial rank at most one or is nuclear purely infinite simple, we provide some additional injectivity conditions to get the above conclusion.

Given a matrix $A$, let $\mathcal{O}\left( A \right)$ denote the orbit of $A$ under a certain group action such as

1. (1) $U\left( m \right)\,\otimes U\left( n \right)$ acting on $m\,\times \,n$ complex matrices $A$ by $(U,\,V)\,*\,A\,=\,UA{{V}^{t}}$,

2. (2) $O\left( m \right)\otimes O\left( n \right)$ or $\text{SO(}m\text{)}\,\otimes \,\text{SO(}n\text{)}$ acting on $m\,\times \,n$ real matrices $A$ by $(U,\,V)\,*\,A\,=\,UA{{V}^{t}}$,

3. (3) $U(n)$ acting on $n\,\times \,n$ complex symmetric or skew-symmetric matrices $A$ by $U\,*\,A\,=\,UA{{U}^{t}}$,

4. (4) $O(n)$ or $\text{SO(n)}$ acting on $n\,\times \,n$ real symmetric or skew-symmetric matrices $A$ by $U\,*\,A\,=\,UA{{U}^{t}}$.

Denote by

1

$$\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})\,=\,\{{{X}_{1\,}}+\cdot \cdot \cdot +\,{{X}_{k}}\,:\,{{X}_{i}}\,\in \,\mathcal{O}({{A}_{i}}),i\,=\,1,\ldots ,k\}$$

the joint orbit of the matrices ${{A}_{1}},\ldots ,{{A}_{k}}$. We study the set of diagonals or partial diagonals of matrices in $\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})$, i.e., the set of vectors $({{d}_{1}},\ldots {{d}_{r}})$ whose entries lie in the $(1,\,{{j}_{1}}),\ldots ,(r,\,{{j}_{r}})$ positions of a matrix in $\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})$ for some distinct column indices ${{j}_{1}},\ldots ,{{j}_{r}}$. In many cases, complete description of these sets is given in terms of the inequalities involving the singular values of ${{A}_{1}},\ldots ,{{A}_{k}}$. We also characterize those extreme matrices for which the equality cases hold. Furthermore, some convexity properties of the joint orbits are considered. These extend many classical results on matrix inequalities, and answer some questions by Miranda. Related results on the joint orbit $\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})$ of complex Hermitian matrices under the action of unitary similarities are also discussed.