Matrix with Functions as Entries

Question

What do we call a matrix with functions as entries?
$$\textbf{f(x)}=\begin{bmatrix} f_{11}(x) & f_{12}(x) \\ f_{21}(x) & f_{22}(x) \end{bmatrix} $$

Answer 1

Recall that you denote by $M_{2\times 2}(\mathbb{C})$ the set of matrices with entries in the complex numbers.

You can define matrices over other sets and depending on the structure of those sets these matrix algebras may or may not be interesting to you.

It is not clear what type of functions the $f_{ij}$ are here but because you might want to add and multiply the entries (why?), they will usually form a ring (briefly a set with addition and multiplication). So perhaps your functions come from the ring of continuous functions on the real line. You might denote this set by $\mathcal{C}(\mathbb{R})$.

Then your matrix above would be an element of the set of matrices over $\mathcal{C}(\mathbb{R})$ which we would denote by $M_{2\times 2}(\mathcal{C}(\mathbb{R}))$ and we could write $$\mathbf{f}=\left(\begin{array}{cc}f_{11} & f_{12}\\ f_{21} & f_{22}\end{array}\right).$$

Your object above then seems to do more. It seems to take as an input $x\in\mathbb{R}$ and output a 2$\times$2 matrix:

$$\mathbf{f}:x\mapsto \left(\begin{array}{cc}f_{11}(x) & f_{12}(x)\\ f_{21}(x) & f_{22}(x)\end{array}\right),$$

so you have a function

$$\mathbf{f}:\mathbb{R}\rightarrow M_{2\times 2}(\mathbb{R}).$$

EDIT: i.e. what copperhat said.

Now you might begin to ask what kind of properties does this map have?

Answer 2

an example: such matrices describe the local behaviour when all the entries are partial derivatives of a function with more than one variable...

if the entries are the first derivative then we call the matrix a "jacobian" matrix Jacobian Wikipedia]1