- #1

MxwllsPersuasns

- 101

- 0

## Homework Statement

**Please bear with the length of this post, I'm taking it one step at a time starting with i)**

Let A: I → gl(n, R) be a smooth function where I ⊂ R is an interval and gl(n, R) denotes the vector space of all n × n matrices.

(i) If F : I → gl(n, R) satisfies the matrix ODE F' = F A , then det F satisfies the scalar ODE (det F)' = tr A det F. Here tr B denotes the trace (the sum of the diagonal elements) of an n × n matrix B.

(ii) If F : I → gl(n, R) satisfies the matrix ODE F' = F A and for some t

_{0}∈ I we have F(t

_{0}) ∈ GL(n, R), where GL(n, R) denotes the group of invertible matrices, then F : I → GL(n, R).

(iii) If F : I → gl(n, R) satisfies the matrix ODE F' = F A and tr A = 0 then det F(t) is constant in t. In particular, if det F(t

_{0}) = 1 then det F(t) = 1 for all t ∈ I.

(iv) If F : I → gl(n, R) satisfies the matrix ODE F' = F A and A: I → SO(n, R) takes values in skew-symmetric matrices, then F

^{T}(t)F(t) is constant in t. In particular, if F(t

_{0}) ∈ O(n, R) then F : I → O(n, R). The analogous statement holds for F(t

_{0}) ∈ SO(n, R).

## Homework Equations

Not sure; I guess the (scalar) derivative of a matrix would equal that matrix with derivatives wrt to that scalar of each element of the original matrix would be something to use.

## The Attempt at a Solution

So I'm guessing the way this is worded that basically we just need to demonstrate each of these facts. So my attempt for part i) was started by assuming we are taking the derivative of the matrix F with respect to some variable (I used t). I took F and A to each be 2 x 2 matrices and had the standard ƒ

_{11}in the upper left corner for F and following through to α

_{22}in the right hand bottom corner for A. Thus each element of A is of the form α

_{ij}(t) and each element of F is of the form ƒ

_{ij}(t) with each being parameterized by the variable t and (i,j) ∈ {1,2}.

So where I'm running into issues is that I'm not sure if; 1) I'm supposed to show the first ODE by differentiating F and showing it equals the original matrix multiplied by another (A), in which case I'm stuck since differentiating F only yields the same matrix but every element is differentiated.. it's not really like just multiplying by another matrix. Or if 2) I'm supposed to solve the matrix ODE in an analogous way to the way that a regular ODE (f'(x) = γf(x)) in which case I'm not sure how exactly I do that? Do I just treat them as a system of two equations (right? 1 for each row) in one variable and solve that way?

Any help along this journey of ODE discovery is immensely appreciated!