# 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)=eλt*x.

Would du/dt be considered matrix A? Thanks!

tatianaiistb said:

## 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)=eλt*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.

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.

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

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?

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

I like Serena said:
[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.