2. For a starting vector of p = [10, 10], compute p^(10) = A^10p for each matrix A in Exercise 1

2. For a starting vector of p = [10, 10], compute p^(10) =...

Transcript

00:01 Hello, we are given a matrix a and we have that a times 3 2 vector is equal to 6 4 vector in a times the vector 1 1 is equal to 1 1.

00:16 So, if we call this vector 3 2 u and the vector 1 1 v, we have that a u is equal to 2 u, a v equal to v.

00:24 So, u and v are eigenvectors of a with eigenvalue 2 and 1 respectively.

00:31 This is part a.

00:33 Now, let us call the operator phi which is represent whose matrix with respect to the standard basis is a.

00:42 Then, ok this is one net and now let us notice that u and v these are not multiples of each other.

00:51 So, they are linearly independent.

00:53 So, u and v are linearly independent.

00:55 So, u and v let us call that set b, they form a basis of r2.

01:02 Now, the matrix p which let us call this matrix p, let us denote by p the matrix which gives the change of basis matrix from the basis b to basis e which changes coordinates in terms of basis b to coordinates in terms of the basis e.

01:24 Now, this matrix will be 3 2 1 1 and this matrix to change from e coordinates to b coordinates will be the inverse of this matrix.

01:34 Now, the inverse of a 2 by 2 matrix is 1 over the determinant.

01:38 Determinant of this thing is 3 minus 2 1 1 over 1 times we switch or swap the two diagonal entries 2 1 3 and take negative of two this these two entries minus 1 minus 2.

01:54 So, the inverse of p which is this thing is 1 minus 1 minus 2 3.

02:03 Now, ok now the operator phi on u was 2u, operator phi on v was v.

02:14 So, the matrix of the operator phi with respect to the uv basis is 2 is a diagonal matrix and it is to this diagonal matrix.

02:25 Now, the matrix of a is the matrix of the operator with respect to e.

02:30 It is equal to the change of basis is it is equal to the change of basis from e to b...

Question

2. For a starting vector of p = [10, 10], compute $p^{(10)} = A^{10}p$ for each matrix A in Exercise 1. $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$

Question

2. For a starting vector of p = [10, 10], compute $p^{(10)} = A^{10}p$ for each matrix A in Exercise 1. $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$

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