Problem 3 a let b beginpmatrix 0 1 1 0 endpmatrix evaluate...

Problem 3. (a) Let $B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. Evaluate $e^{Bc}$, where $c \in \mathbb{R}$ is a constant. (b) Let $A = \begin{pmatrix} a & b \\ -b & a \end{pmatrix} = aI + bB$. Evaluate $e^{At}$. (c) Use the matrix exponential to solve the system $\vec{x}' = A\vec{x}$. (d) Use the matrix exponential to solve the system $\vec{x}' = A\vec{x} - \begin{pmatrix} b \\ a \end{pmatrix}$.

Transcript

00:01 In this video, we're going to go through the answer to question number 17 from chapter 9 .5.

00:05 So we're given a matrix a, and we're told to consider the system x prime is equal to ax.

00:19 So a is 2x2 matrix, and we're asked to do a series of, to answer a series of questions about this system.

00:31 So in part a, we're asked to verify the eigenvalues and iconvector.

00:34 So we don't actually need to derive them, we just need to verify that they are indeed the icon vectors and icon values of this matrix.

00:43 So all we need to do is show that a minus r1 times i, times the identity matrix times by the first icon vector is zero.

01:02 Okay, so let's do that.

01:03 So a minus r1, which is 2, times the identity matrix, is minus 1, root 3, root 3, minus 3 times by u1, which is the column vector root 3, 1.

01:30 So this is going to be the top component, it's going to be minus 3 plus root 3, which is 0, and root 3, and root 3, which is 0.

01:41 Times root 3, which is 3 minus 3, which is 0.

01:46 So at least r1 and u1 are an eigenvector, an eigenvalue and eigenvector.

01:56 Let's try and show the same for r2 and u2.

02:08 R2.

02:08 R2 is minus 2.

02:10 So 1 minus minus 2 is 3.

02:15 Then we've got root 3, root 3 minus 1 plus 2 is 1, and we're told that the eigenvector is 1 minus root 3.

02:39 Okay, so 3 minus root 3 times root root 3, which is 3 minus 3 is 0, and root 3 minus root 3 is 0.

02:52 So yeah, r2 and u2 are also an icon value and icon vector.

03:00 It's a 2x2 matrix, so these can be the only icon values and icon vectors of the matrix.

03:07 Okay, part b.

03:13 So it's going to be helpful now, before we get ahead with the sketching of the trajectories, it's going to be helpful to write down the general solution of this equation, which we can do because we have a linear system, a linearly independent system of eigenvectors and the corresponding icon values.

03:40 So it's going to be a constant c1 times by e to the first eigenvalue, which is 2 times t, times by the eigenvector, which is root 3 ,1, plus a second constant times times e to the second item value minus 2, t times the second icon vector 1 minus root 3.

04:20 Okay, so we'll have that in the back of our mind.

04:22 So that has to sketch the trajectory of solution with initial vector minus u1.

04:29 Okay, so this is our grid.

04:36 So what does minus u1 look like? so u1 is this guy.

04:44 So minus that is going to be minus root 3...

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