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I'm pretty clear in my understanding of scalar-valued differential $(p, q)$-forms (resp. holomorphic $(p,0)$-forms) on a complex manifold $M$ and the related Hodge theory. What I'm not sure about is whether there is a matrix-valued analogue of such differential forms on a complex manifold in the math literature. If there is such a thing, what are some good references that I may find useful?

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    $\begingroup$ A matrix-valued differential form is just a matrix of scalar differential forms. Writing it as a matrix is just a convenient way to do calculations. For example, you can check easily that exterior multiplication and matrix multiplication combine nicely into exterior matrix multiplication, as long as you don't change the order of the factors. $\endgroup$
    – Deane Yang
    Commented Apr 14, 2018 at 0:11

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Actually, matrix-valued differential forms are used a lot in hypercomplex analysis/hypercomplex geometry, which, as the name suggests, includes certain complex manifolds. There is a nice account of such differential forms in Rocha-Chavez, Shapiro, and Sommen "Integral Theorems for Functions and Differential Forms in $\mathbb{C}^m$" (2001) (~200 pages). The first chapter is a general introduction to (complex-valued) differential forms, the second chapter discusses differential forms with values in the 2x2-matrices, and then the book goes on into the realm of hyperholomorphy, including hyperholomorphic Hodge theory.

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Chern used matrix-valued differential forms frequently, both on real and complex manifolds. For example, S. S. Chern, Complex manifolds without potential theory, Van Nostrand, 1967, p. 34.

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