Let A = eginpmatrix 3 1 1 1 0 2 1 2 ...

Let A = egin{pmatrix} 3 & 1 & 1 \ 1 & 0 & 2 \ 1 & 2 & 0 end{pmatrix} a. Find the normalized eigenvectors and use them as columns in an orthogonal matrix C. b. Show that C'AC = D, where D is diagonal with the eigenvalues of A on the diagonal. c. Show that A = CDC'.

Transcript

00:03 So, the given matrix here is the given matrix a is equal to 3 1 1 1 0 2 and 1 2 0.

00:24 Now, x is equal to x 1 x 2 x 3 be the corresponding corresponding eigenvector eigenvector.

00:58 What we have is that a x is equal to lambda into x.

01:12 So, implies that a minus lambda into i into x is equal to 0.

01:28 Here let us consider first lambda is equal to 1.

01:33 So, what we are getting here then 3 minus lambda 1 1 1 0 minus lambda 2 1 2 and 0 minus lambda into x 1 x 2 x 3 is coming as 0 0 0.

02:06 So, now what we have is that to if we solve it for lambda is equal to 1 then it is coming as 2 1 1 1 minus 1 2 1 2 minus 1 into x 1 x 2 x 3 is coming as is equal to 0 0 0.

02:44 So, from there if we solve the equation then we can easily find that x 1 x 2 x 3 is coming as 3 minus 3 minus 3.

02:59 So, for lambda is equal to 1 lambda is equal to 1 the eigenvalue eigenvector x is equal to 3 minus 3 and minus 3.

03:17 Now, if we similarly find the eigenvector for lambda is equal to 4 then we will get the eigenvector as x is equal to 6 3 3 and for lambda is equal to minus 2 we will get the eigenvector x is equal to 0 minus 9 9.

03:56 So, now if we then calculate c or the orthogonal matrix then the c or orthogonal matrix will come as 3 by root over 27 minus 3 by root over 27 minus 3 by root over 27 6 by root over 54 3 by root over 54 3 by root over 54 0 by root over 162 minus 9 by root over 162 and 9 by root over 162.

04:52 So, this is our orthogonal matrix.

04:56 Now, is the answer for question a now if we go to question b then the d is given as 1 0 0 0 4 0 0 minus 2 we are asked to find c transpose a c is equal to d.

05:28 So, if we first find c transpose into a then that will be is equal to 3 by root over 27 6 by root over 54 0 and minus 3 by root over 27 12 by root over 54 18 by root over 162 and minus 3 by root over 27 12 by root over 54 and minus 18 by root over 162...

Question

Let A = egin{pmatrix} 3 & 1 & 1 \ 1 & 0 & 2 \ 1 & 2 & 0 end{pmatrix} a. Find the normalized eigenvectors and use them as columns in an orthogonal matrix C. b. Show that C'AC = D, where D is diagonal with the eigenvalues of A on the diagonal. c. Show that A = CDC'.

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Let A = egin{pmatrix} 3 & 1 & 1 \ 1 & 0 & 2 \ 1 & 2 & 0 end{pmatrix} a. Find the normalized eigenvectors and use them as columns in an orthogonal matrix C. b. Show that C'AC = D, where D is diagonal with the eigenvalues of A on the diagonal. c. Show that A = CDC'.