Diagonalization of a matrix with repeated eigenvalues

In summary, the conversation discusses the concept of diagonalizing a matrix with repeated eigenvalues. The participants suggest trying a simple example, such as the identity matrix, to understand the process. They also mention that to find eigenvalues, one must solve the equation A-\lambda I=0 and not assume that the eigenvalue is already known.
  • #1
bemigh
30
0
Hey guys,
I know its possible to diagonalize a matrix that has repeated eigenvalues, but how is it done? Do you simply just have two identical eigenvectors??
Cheers
Brent
 
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  • #2
Well, try an example -- the simplest you can think of. How about the identity matrix? Can you diagonalize it? What are its eigenvectors?
 
  • #3
Ok, well... i would get an eigenvalue of 1, with a multiplicity of 3. Alright, so when solving for my eigenvalues, by plugging 1 into my matrix A-(lamda)I ; i would just be left with 0's. How can i get eigenvalues from this?? I appreciate your help by the way...
 
  • #4
Am I misunderstanding something? If you know what eigenvalues are, then you must know how to find them in this simple case. If [itex]\lambda[/itex] is an eigenvalue of matrix A, then [itex]A-\lambda I[/itex] is NOT invertible (since the equation A- \lambda I= 0 does not have a unique solution) and so det(A- \lambda I)= 0. Since you are finding the eigenvalues, you do NOT yet know that [itex]\lambda= 1[/itex] so you are NOT "left with 0's. In this case, that equation is [itex](1- \lamba)^3= 0[/itex] which obviously has 1 as a triple root.
 

What is diagonalization of a matrix with repeated eigenvalues?

Diagonalization of a matrix with repeated eigenvalues is the process of finding a diagonal matrix that is similar to the original matrix, using eigenvectors. This is done by finding a basis of eigenvectors, which are vectors that do not change direction during a linear transformation, and using them to construct a new matrix that is in diagonal form.

Why is diagonalization important?

Diagonalization is important because it simplifies the computation and analysis of a matrix. A diagonal matrix is easier to work with than a non-diagonal matrix, as it has many properties that are useful in various applications, such as computing powers of a matrix, solving systems of linear equations, and finding the inverse of a matrix.

Can a matrix with repeated eigenvalues always be diagonalized?

No, not all matrices with repeated eigenvalues can be diagonalized. For a matrix to be diagonalizable, it must have a complete set of linearly independent eigenvectors. If a matrix has repeated eigenvalues, it may not have enough linearly independent eigenvectors to form a basis, and therefore cannot be diagonalized.

What happens if a matrix with repeated eigenvalues cannot be diagonalized?

If a matrix with repeated eigenvalues cannot be diagonalized, it is said to be defective. This means that the matrix does not have a complete set of linearly independent eigenvectors and cannot be simplified to a diagonal form. In this case, other methods such as Jordan decomposition may be used to analyze the matrix.

How can I tell if a matrix has repeated eigenvalues?

A matrix has repeated eigenvalues if the characteristic polynomial of the matrix has a repeated root. This can be determined by finding the eigenvalues of the matrix and checking if any of them have a multiplicity greater than 1. Alternatively, the trace (sum of the diagonal elements) and determinant of the matrix can also give clues about the eigenvalues and their multiplicities.

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