How Do Matrix ODEs Relate to Determinants and Traces?

In summary: 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 ƒ11 in the upper left corner for F and following through to α22 in the right hand bottom corner for A. Thus each element of A is of the form αij(t) and each element of F is of the form ƒij(t) with each being parameterized by the variable t and (i,j) ∈ {1,2}.So where
  • #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 t0 ∈ I we have F(t0) ∈ 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(t0) = 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 FT(t)F(t) is constant in t. In particular, if F(t0) ∈ O(n, R) then F : I → O(n, R). The analogous statement holds for F(t0) ∈ 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 ƒ11 in the upper left corner for F and following through to α22 in the right hand bottom corner for A. Thus each element of A is of the form αij(t) and each element of F is of the form ƒij(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!
 
Physics news on Phys.org
  • #2
MxwllsPersuasns said:

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 t0 ∈ I we have F(t0) ∈ 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(t0) = 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 FT(t)F(t) is constant in t. In particular, if F(t0) ∈ O(n, R) then F : I → O(n, R). The analogous statement holds for F(t0) ∈ 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 ƒ11 in the upper left corner for F and following through to α22 in the right hand bottom corner for A. Thus each element of A is of the form αij(t) and each element of F is of the form ƒij(t) with each being parameterized by the variable t and (i,j) ∈ {1,2}.
Actually seeing the matrices involved would be helpful. In the 2x2 case you have this for the matrix F(t):
##F(t) = \begin{bmatrix}f_{11}(t) & f_{12}(t) \\ f_{21}(t) & f_{22}(t) \end{bmatrix}##
and similarly for A(t):
##A(t) = \begin{bmatrix}a_{11}(t) & a_{12}(t) \\ a_{21}(t) & a_{22}(t) \end{bmatrix}##
For part i), you're given that F satisfies the equation F' = F A. Can you show that |F|' equals tr(A) |F|?
MxwllsPersuasns said:
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!
 
  • #3
Okay so basically what I did was multiply F and A (I won't write out the matrix here) then take Det(FA) (since that's det(F)') and I ended up getting (after some cancellations) something like (and I won't use subscripts here as it would take too long so please forgive me)

Det(FA) = {f11*α11*f22*α22 + f12*α21*f21*α12} - {f11*α12*f22*α21 + f12*α22*f21*α11}

and so then I try calculating det(F)*tr(A)...

|F|*tr(A) = {f11*f22 - f12*f21}*(α11 + α22} but here I can already see that I'll get terms with only 3 things being multiplied (rather than four per term in the Det(FA)) I'm not sure if maybe I did something wrong with my computations (I doubled-checked and feel confident) or perhaps I'm missing some cancellation somewhere but I just don't see how to get these two things to equal one another. Any suggestions?
 
  • #4
Also for number ii) is this basically saying that if there's a point t_0 for which F belongs to the group of invertible matrices (and thus is invertible at that point) and it satisfies F' = FA then F is an invertible matrix? I'm not sure how I could argue that. Or if I'm totally off base here can someone please assist?
 

Related to How Do Matrix ODEs Relate to Determinants and Traces?

1. What is a linear ODE?

A linear ODE (ordinary differential equation) is a mathematical equation that involves a function and its derivatives. It can be written in the form of y' + p(x)y = g(x), where y' represents the derivative of y with respect to x, and p(x) and g(x) are functions of x.

2. How do you prove basic linear ODE results?

The proofs of basic linear ODE results involve using techniques from calculus, such as integration and differentiation, along with the properties of linear equations. It is also important to have a strong understanding of the fundamental concepts of ODEs and their solutions.

3. What are some common results for linear ODEs?

Some common results for linear ODEs include the existence and uniqueness theorem, which states that there is a unique solution to a first-order linear ODE given certain initial conditions, and the superposition principle, which states that the sum of two solutions to a linear ODE is also a solution.

4. How are linear ODEs used in scientific research?

Linear ODEs are used in a variety of scientific fields, such as physics, engineering, and biology, to model and analyze various physical systems. They can also be used to solve problems in economics, population dynamics, and other areas of mathematics.

5. What are some real-world applications of linear ODEs?

Linear ODEs have numerous real-world applications, including predicting the growth of bacteria in a petri dish, modeling the trajectory of a projectile, and determining the rate of change in a chemical reaction. They are also used in control theory to design systems that can respond to changing conditions.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
623
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
872
  • Calculus and Beyond Homework Help
Replies
5
Views
10K
  • Calculus and Beyond Homework Help
Replies
11
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Replies
1
Views
3K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
Back
Top