I have the matrix
-7 -5 -7
10 11 10
-7 -5 -7
I need to find the characteristic equation, and find three eigenvectors and eigenvalues.
Could you please show your working as I'm trying to learn :)
-7 -5 -7
10 11 10
-7 -5 -7
I need to find the characteristic equation, and find three eigenvectors and eigenvalues.
Could you please show your working as I'm trying to learn :)
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Firstly, an eigenvalue, b, is any solution to the problem Ax=bx, where b is a scalar, x a vector, and A is your original 3x3 matrix. We find b by taking the determinant of A-bI, where I is the identity matrix. The resulting matrix, A-bI is:
(-7-b) (-5) (-7)
(10) (11-b) (10)
(-7) (-5) (-7-b)
Take the determinant and set it to zero. It should be a cubic for b. You will then have three solutions for b, although they may not be unique. They are your three eigenvalues. I get -9, 6, and 0.
To find the vectors, plug in the specific values for b into the equation Ax=bx and solve for the values of x1, x2, and x3. That is your first eigenvector. Do it again for your second and third eigenvalues.
(-7-b) (-5) (-7)
(10) (11-b) (10)
(-7) (-5) (-7-b)
Take the determinant and set it to zero. It should be a cubic for b. You will then have three solutions for b, although they may not be unique. They are your three eigenvalues. I get -9, 6, and 0.
To find the vectors, plug in the specific values for b into the equation Ax=bx and solve for the values of x1, x2, and x3. That is your first eigenvector. Do it again for your second and third eigenvalues.