27 99 92 now let a 29 531 and find a symmetric matrix p and a skew symmetric matrix such that a

Name: 27 99 92 now let a 29 531 and find a symmetric matrix p and a skew symmetric matrix such that a pq 69 44 67 p o sin 0 82 sin 82 0 0 0 0 0 0 29765
Uploaded: 2022-11-25T13:25:03-08:00
Duration: 6 min 35 s
Channel: Adi S
Description: 27 99 92 now let a 29 531 and find a symmetric matrix p and a skew symmetric matrix such that a pq 69 44 67 p o sin 0 82 sin 82 0 0 0 0 0 0 29765

First, we need to find the matrix Q, which is skew-symmetric. This means that Q^T = -Q, where Q^

27 99 92 now let a 29 531 and find a symmetric matrix p and...

Transcript

00:01 In this problem, we are given the symmetric matrix s to be the matrix 1 -02 -0 -1 -2 -2 -2 -2 -2 -2 -0 -negative 2 -0.

00:15 We are asked to find the orthogonal matrix q that will diagonalize this matrix s.

00:25 To find the orthogonal matrix q, first we need to find the eigenvalues and the corresponding.

00:31 Eigenvectors of this symmetric matrix s so the eigen values are obtained by solving the equation determinant of s minus lambda i is equal to zero where lambda is the eigen value and i is the three by three identity matrix so this is determinant of one minus lambda zero to zero negative one minus lambda negative to two two negative to negative to negative lambda the determinant of this is equal to zero.

01:05 Sol this equation we have this is 1 minus lambda times negative 1 minus lambda times times negative lambda minus 4 plus 2 times negative 1 minus lambda is equal to 0 from which we can simplify this as 1 minus lambda times 1 minus lambda times.

01:32 Lambda times 1 plus lambda minus 4 plus 4 times 1 plus lambda is equal to 0, which further simplifies to lambda times 1 minus lambda square minus 4 plus 4 lambda is equal to 0 from which we get the equation lambda cube minus 9 minus 9.

02:04 9 lambda is equal to 0 this can be factized as lambda times lambda square minus 9 is equal to 0 by 0 product property we get the eigen values are lambda is equal to 0 or lambda square is equal to 9 that is the eigen values are lambda 1 is equal to 0 lambda 2 is equal to 3 and lambda 3 is equal to negative 3 now corresponding to each eigen value we need to find the eigen vector so consider lambda 1 is equal to 0.

02:39 Then to get the corresponding eigenvector, we need to solve the system s minus lambda 1 i times x is equal to 0 where x must be non -0.

02:50 So that means we must solve the matrix equation 1 -02, 0 -negative 1 -2, 2 -negative 2 -0 times x1, x2 -2 -6 -3 is equal to the 0 -metrics 0 -0 -0 -0 -0 -0.

03:06 So this can be also simplified as the matrix 1 -02, 0 -1 -negative 2, 0 -negative 2 -0 -2 -2 -negative 4 times x -1, x -2 -3 is equal to the 0 -3 matrix 0 -000...

Question

27 99 92 Now let A = 29 531 and find a symmetric matrix P and a skew-symmetric matrix such that A = P+Q 69 44 67 P = 'O- sin (0) 82 sin (@) 82 0-0-0 0-0-0

Question

27 99 92 Now let A = 29 531 and find a symmetric matrix P and a skew-symmetric matrix such that A = P+Q 69 44 67 P = 'O- sin (0) 82 sin (@) 82 0-0-0 0-0-0

Please give Ace some feedback