Solve du/dt=Pu when P is a projection

[tatianaiistb]

Homework Statement

Solve du/dt=Pu when P is a projection.

[1/2 1/2 = du/dt with
1/2 1/2]

[5 = u(0).
3]

Part of u(0) increases exponentially while the nullspace part stays fixed.

Homework Equations

du/dt = Au with u=u(0) at t=0

The Attempt at a Solution

I'm stuck with this problem. I was thinking that u(t)=e^λt*x.

Would du/dt be considered matrix A? Thanks!

 

[Mark44]

Homework Statement

Solve du/dt=Pu when P is a projection.

[1/2 1/2 = du/dt with
1/2 1/2]

[5 = u(0).
3]

Part of u(0) increases exponentially while the nullspace part stays fixed.

Homework Equations

du/dt = Au with u=u(0) at t=0

The Attempt at a Solution

I'm stuck with this problem. I was thinking that u(t)=e^λt*x.

Would du/dt be considered matrix A? Thanks!

I don't understand much of what you have written.
Is this P?
[1/2 1/2]
[1/2 1/2]

If so, it's not a projection matrix - it is just 1/2 I, the identity matrix.

du/dt is a two-dimensional vector, whose components are du₁/dt and du₂/dt.

Your system of differential equations, not written in matrix form would look something like this:

du₁/dt = a₁₁u₁ + a₁₂u₂
du₂/dt = a₂₁u₁ + a₂₂u₂

This could be written in vector/matrix form as du/dt = Au.

 

[tatianaiistb]

That's where I'm confused at.

[1/2 1/2 = du/dt. So I guess this means that the matrix equals Pu.
1/2 1/2]'

I'm not sure what to make of the matrix.

 

[Mark44]

That doesn't make any sense to me. What is the exact wording of the problem?

 

[I like Serena]

Hi tatianaiistb!

[1/2 1/2]
[1/2 1/2]
is the matrix for the orthogonal projection on (1,1).
(Sorry Mark )

So with u(t)=(x(t), y(t)) the problem reads:
$$\begin{align}\dot x ={1 \over 2}x + {1 \over 2}y \\ \dot y ={1 \over 2}x + {1 \over 2}y\end{align}$$
with
$$\begin{align}x_0 = 5 \\ y_0 = 3\end{align}$$

Can you solve that?

 

[tatianaiistb]

Thank you both... I think I figured out the problem. I greatly appreciate it!

 

[Mark44]

[1/2 1/2]
[1/2 1/2]
is the matrix for the orthogonal projection on (1,1).
(Sorry Mark )

No problem and no need for apology. I wasn't able to make heads or tails out of what she was trying to do, so I'm glad you jumped in.