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Let $g:\mathbb{C} \mapsto \mathbb{C}^{n \times n}$. How does one define the integral $$\int_\Gamma g(z) dz,$$ for some closed curve $\Gamma \subset \mathbb{C}$?

Is it understood entry-wise or as an special case of the Bochner Integral? I'm trying to understand the Matrix Cauchy integral formula for the definition of matrix functions.

$$f(A) = \frac{1}{2\pi i}\int_\Gamma f(z)(z I - A)^{-1}dz$$ where $f$ is analytic on and inside a closed contour $\Gamma$ that encloses $\Lambda(A)$.

Duncan Idaho
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    The matrix integral is typically just defined component-wise. By linearity, if A is diagonalizable you can move to a basis where A is diagonal and then apply the integral to each of the eigenvalues. – overfull hbox Jan 18 '24 at 02:51
  • Thanks @overfullhbox, do you by any chance have a reference for this? – Duncan Idaho Jan 19 '24 at 19:00
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    Not sure, here is another SE post though: https://math.stackexchange.com/questions/2242369/integrate-int-ax-dx-where-ax-beginpmatrixf-11xf-12x-f-21?noredirect=1&lq=1 – overfull hbox Jan 20 '24 at 13:24

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