Let A=(A1,…,Am)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{A}= (A_1, \ldots , A_m)$$\end{document}, where A1,…,Am\documentclass[12pt]{minimal}
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\begin{document}$$A_1, \ldots , A_m$$\end{document} are n×n\documentclass[12pt]{minimal}
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\begin{document}$$n\times n$$\end{document} real matrices. The real joint (p, q)-matricial range of A\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{A}$$\end{document}, Λp,qR(A)\documentclass[12pt]{minimal}
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\begin{document}$${\varLambda }^{{\mathbb {R}}}_{p,q}(\mathbf{A})$$\end{document}, is the set of m-tuple of q×q\documentclass[12pt]{minimal}
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\begin{document}$$q\times q$$\end{document} real matrices (B1,…,Bm)\documentclass[12pt]{minimal}
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\begin{document}$$(B_1, \ldots , B_m)$$\end{document} such that (X∗A1X,…,X∗AmX)=(Ip⊗B1,…,Ip⊗Bm)\documentclass[12pt]{minimal}
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\begin{document}$$(X^*A_1X, \ldots , X^*A_mX) = (I_p\otimes B_1, \ldots , I_p \otimes B_m)$$\end{document} for some real n×pq\documentclass[12pt]{minimal}
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\begin{document}$$n \times pq$$\end{document} matrix X satisfying X∗X=Ipq\documentclass[12pt]{minimal}
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\begin{document}$$X^*X = I_{pq}$$\end{document}. It is shown that if n is sufficiently large, then the set Λp,qR(A)\documentclass[12pt]{minimal}
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\begin{document}$${\varLambda }^{{\mathbb {R}}}_{p,q}(\mathbf{A})$$\end{document} is non-empty and star-shaped. The result is extended to bounded linear operators acting on a real Hilbert space H\documentclass[12pt]{minimal}
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\begin{document}$${{{\mathcal {H}}}}$$\end{document}, and used to show that the joint essential (p, q)-matricial range of A\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{A}$$\end{document} is always compact, convex, and non-empty. Similar results for the joint congruence matricial ranges on complex operators are also obtained.
