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Math Help - Eigenvalues, Eigenvectors and Diagonalization

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    Eigenvalues, Eigenvectors and Diagonalization

    I have the matrix A=|7 2|
    |12 -3|
    I found the eigenvalues λ=9 and λ=-5
    Now i have to find the eigenvector, using Gauss.
    I set
    A-9I = |-2 2 0| R1
    |12 -12 0| R2
    and used Gauss this way:
    R2+5*R1
    R1/-2 and R2/2
    R2/R1
    R2-R1
    R2/2
    R1+R2
    until i got the vector
    |1 0 0|
    |0 1 0|
    I cant really find out what to do now. And then i have to find a matrix, P, that diagonalizes the matrix A.


    In the other example i had the vector
    A= |-1 -2 -2|
    |1 2 1|
    |-1 -1 0|
    and i solved for the eigenvalues and got
    -λ^3+λ^2-3λ-1
    But if i solve it with my texas instrument, i get λ=-0,295598 wich doesn't sound so right. So how do i solve it?

    Thank you
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  2. #2
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    Quote Originally Posted by laugalitla View Post
    I have the matrix A=|7 2|
    |12 -3|
    I found the eigenvalues λ=9 and λ=-5
    Now i have to find the eigenvector, using Gauss.
    I set
    A-9I = |-2 2 0| R1
    |12 -12 0| R2
    and used Gauss this way:
    R2+5*R1
    Okay, that would give \left[\begin{array}{ccc}-2 & 2 & 0 \\ 2 & -2& 0 \end{array}\right]

    R1/-2 and R2/2
    and this will give you \left[\begin{array}{ccc}1 & -1 & 0\\ 1 & -1 & 0\end{array}\right]

    R2/R1
    I don't know what you mean by this. Dividing one row by another is not a "row-operation".

    R2-R1
    If I ignore the previous line and subtract R1- R2, I get \left[\begin{array}{ccc}1 & 1 & 0 \\ 0 & 0 & 0\end{array}\right] and I can stop here. An eigenvector is \left[\begin{array}{c} 1 \\ 1 \end{array}\right].

    R2/2
    R1+R2
    until i got the vector
    |1 0 0|
    |0 1 0|
    First, that is not a vector, it is a matrix. Also you should have realized that there was something wrong. 9 was an eigenvalue because det(A- 9I)= 0 which means that A is not invertible and cannot be row reduced to the identity matrix as you have.

    [quot]I cant really find out what to do now. And then i have to find a matrix, P, that diagonalizes the matrix A.[/quote]

    A more fundamental way of doing this: from the definition of "eigevalue" there exist a non-zero vector v such that \left[\begin{array}{cc}7 & 2 \\ 12 & -3\end{array}\right]\left[\begin{array}{c}x \\ y\end{array}\right]= 9\left[\begin{array}{c}x \\ y\end{array}.
    Multiplying that out gives the three equations 7x+ 2y= 9x, 12x- 3y= 9y or 2x+ 2y= 0 and 3x- 3y= 0 both of which are satified by any x,y such that y= x. Again, that says that \left[\begin{array}{c} 1 \\ 1 \end{array}\right] is an eigenvector.


    In the other example i had the vector
    A= |-1 -2 -2|
    |1 2 1|
    |-1 -1 0|
    and i solved for the eigenvalues and got
    -λ^3+λ^2-3λ-1
    But if i solve it with my texas instrument, i get λ=-0,295598 wich doesn't sound so right. So how do i solve it?

    Thank you
    You don't show how you got that equation but that's not what I get.
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