Abstract Matrix Algebra and Calculus in sympy

Question

I am doing control engineering and I often face problems of the type below and I want to know if there is a way to deal with this in sympy.

question: tl:dr: I want to make a MatrixSymbol dependent on a scalar Symbol representing time, to allow differentiation w.r.t. time.

Actual problem: v(t)=[v1(t),v2(t),v3(t)] is a vector function of the time t and I want to calculate the Projection into the direction of v and it's time derivative. In the end I would love to get an expression of v, v.diff(t) and v.T (the transpose).

attempts:

I've tried different things and show the closest one:

This does the algebra I need, but I cannot take derivatives w.r.t. time

v = MatrixSymbol('v',3,1) 

# here i'm building the terms I want
projection_v = v*sqrt(v.T*v).inverse()*v.T
orthogonal_v = Identity(3)-projection_v

orthogonal_v.as_explicit() 

orthogonal_v shows the abstract equation form that I need. In the end - to check and see the result again, I'd also like to make it explicit and see the expression as a function of v[0,0], v[1,0], and v[2,0] for MatrixSymbol the function .as_explicit() does exactly that beginning with sympy version 1.10. (Thanks Francesco Bonazzi for pointing this out.)

The problem however is, that I cannot make these a function of t and take the derivative of projection_v w.r.t. the time t.

I also tried

t = Symbol('t',real=True,positive=True)
v1 = Function('v1',real=True)(t)
v2 = Function('v2',real=True)(t)
v3 = Function('v3',real=True)(t)
v_mat = FunctionMatrix(3,1,[v1,v2,v3]);

but it seems FunctionMatrix is meant to evaluate the functions directly instead of being an analog to the scalar Function.

Effectively I want to be able to calculate orthogonal_v.diff(t) and then see the component wise operations with something like orthogonal_v.diff(t).as_explicit(). Is this possible?