Symbolic diagonalization, matrix differentiation and the wedge product

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The apply_map method is a handy command for entrywise operations. For instance, let

A = matrix([[sin(x), cos(x)],[cos(x), -sin(x)]]);  # define a matrix function of x
A

$$\begin{pmatrix} \sin\left(x\right) & \cos\left(x\right) \\ \cos\left(x\right) & -\sin\left(x\right) \end{pmatrix}.$$

Let's compute the entrywise derivative with apply_map:

A.apply_map(lambda a : a.derivative(x))

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} \cos\left(x\right) & -\sin\left(x\right) \\ -\sin\left(x\right) & -\cos\left(x\right) \end{array}\right),$$

and the Jordan canonical form

A.jordan_form()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{r|r} -\sqrt{\cos\left(x\right)^{2} + \sin\left(x\right)^{2}} & 0 \\ \hline 0 & \sqrt{\cos\left(x\right)^{2} + \sin\left(x\right)^{2}} \end{array}\right).$$

Another small example, this time with symbolic functions:

var('x y')
m=2; n=4

# create a mxn matrix of symbolic functions in x and y
F = matrix([[function('f'+str(i)+str(j))(x, y) for j in [1..n]] for i in [1..m]]); 
F

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} f_{11}\left(x, y\right) & f_{12}\left(x, y\right) & f_{13}\left(x, y\right) & f_{14}\left(x, y\right) \\ f_{21}\left(x, y\right) & f_{22}\left(x, y\right) & f_{23}\left(x, y\right) & f_{24}\left(x, y\right) \end{array}\right).$$

Now, let's ask for the entrywise integral with respect to $y$ of the 2nd derivative with respect to $x$:

F.apply_map(lambda f : (f.derivative(x, 2)).integrate(y))

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} \int \frac{\partial^{2}}{(\partial x)^{2}}f_{11}\left(x, y\right)\,{d y} & \int \frac{\partial^{2}}{(\partial x)^{2}}f_{12}\left(x, y\right)\,{d y} & \int \frac{\partial^{2}}{(\partial x)^{2}}f_{13}\left(x, y\right)\,{d y} & \int \frac{\partial^{2}}{(\partial x)^{2}}f_{14}\left(x, y\right)\,{d y} \\ \int \frac{\partial^{2}}{(\partial x)^{2}}f_{21}\left(x, y\right)\,{d y} & \int \frac{\partial^{2}}{(\partial x)^{2}}f_{22}\left(x, y\right)\,{d y} & \int \frac{\partial^{2}}{(\partial x)^{2}}f_{23}\left(x, y\right)\,{d y} & \int \frac{\partial^{2}}{(\partial x)^{2}}f_{24}\left(x, y\right)\,{d y} \end{array}\right).$$