SMART MATH PEOPLE !! help please.?

Question

Find the possible eigenvalues, eigenvectors and if possible find an invertible matrix P such that P^-1AP=D.

A= 2 -1
-4 -1

The polynomial is x^2 - x - 6 and the eigen values are -2 and 3. The eigenvectors are [1 4]^t and [-1 1]^t where ^t means the transpose of these matrices. I just need help wit the invertible matrix part.. and explanation how to do it would be great.

Answer 1

All you need to do now is put together (concatenate) the column eigenvectors you've found into a 2x2 matrix, take that as P, and take as D the diagonal matrix made up of the eigenvalues (in the same order as the eigenvectors). So P =
1 -1
4 1
and D =
-2 0
0 3
Then P^(-1)AP = D.
[In the other answer, P and its inverse are misplaced, if P is made up of e-vectors.]

Algebraic explanation: P^-1AP = D means the same as AP = PD, and this just says that the columns of P are eigenvectors of A. To see this, look at matrix multiplication column-by-column: look at AP as n products of A times a column of P. IOW, you can get AP by splitting P into columns, premultiplying each by A, and putting the resulting columns together side by side, like this: AP = [A1stColofP, A2ndColofP,...]. Similarly PD = [P1stColofD, P2ndColofD,...]---which equals [1stColofPtimes1stEvalue, 2ndColofPtimes2ndEvalue,...] because, for any column d, Pd is a linear combination of the columns of P, and this boils down to a scalar multiple of a single column of P when d is a column of a diagonal matrix such as D.

Answer 2

You want A = P D P^{-1} The matrix P^{-1} transforms the coordinates in the standard basis into the coordinates in the diagonal basis and conversely P transforms the coordinates in the diagonal basis into the coordinates in the standard basis. Therefore the columns of P are the components of the eigenvectors. So, from what you said P = 1 -1 ( first line)
4 1 Got it?

Answer 3

What i'd do first, is in each and each and every equation, sparkling the fractions by multiply the entire equation by the least problem-free denominator. x/3 + y/4 = 2 i visit multiply by 12. 4x + 3y = 24 2x/3 - y/2 = 0 i visit multiply by 6 4x - 3y = 0 Now you've 2 equations. 4x + 3y = 24 4x - 3y = 0 note that you do not favor to multiply those by anyhting.. the y's cancel out. upload both equations and get: 8x = 24 x = 3 Now replace this lower back in: 4x - 3y = 0 4(3) - 3y = 0 12 = 3y 4 = y the answer is (3 , 4). wish that permits!

Answer 4

http://en.wikipedia.org/wiki/Invertible_matrix