Eigenvalues and Eigenvectors of a Non-Diagonalizable Matrix

In summary, the conversation discusses a matrix with 0, 1, and 0 values and the possibility of it having a diagonal form. The eigenvalues of 0 and 1 are mentioned, along with the system that can be obtained by using 0. It is determined that there is only one linearly independent eigenvector corresponding to the eigenvalue of 0, making the matrix not diagonalizable.
  • #1
Yankel
395
0
Hello,

sorry that I am asking too many questions, I am preparing for an exam...

I have a matrix,

0 1 0
0 0 0
0 0 1

and I need to say if it has a diagonal form (I mean, if there are P and D such that D=P^-1*D*P)

I found that the eigenvalues are 0 and 1. I also know that if I use 0, I get the system

0 1 0 0
0 0 0 0
0 0 0 0

(after Gaussian process)

What can I say about the eigenvectors, do they exist ? the eigenvalue 0 had a dimension of 2. so I need 2 eigenvectors in order to say that P and D exist...
 
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  • #2
There is only one linearly independent eigenvector (1, 0, 0) corresponding to the eigenvalue of 0. So, the matrix is not diagonalizable.
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used to understand the behavior of linear transformations. Eigenvalues represent the scalar values by which an eigenvector is scaled when transformed by the linear transformation. Eigenvectors are the corresponding vectors that are only scaled, but not rotated, by the linear transformation.

2. How are eigenvalues and eigenvectors calculated?

Eigenvalues and eigenvectors are calculated by solving the characteristic equation of a square matrix. The characteristic equation is a polynomial equation formed by subtracting the identity matrix multiplied by a scalar from the original matrix. The roots of this polynomial equation are the eigenvalues, and the corresponding eigenvectors can be found by solving for the null space of the matrix formed by substituting each eigenvalue into the characteristic equation.

3. What is the significance of eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are important in many areas of mathematics and science because they provide a way to simplify complex linear transformations. They allow for the decomposition of a matrix into its basic components, making it easier to analyze and understand the behavior of the transformation. They are also used in many practical applications, such as in image and signal processing.

4. Can a matrix have complex eigenvalues and eigenvectors?

Yes, a matrix can have complex eigenvalues and eigenvectors. This is because the characteristic equation can have complex roots, resulting in complex eigenvalues. The corresponding eigenvectors will also be complex. In fact, for matrices with real entries, the complex eigenvalues and eigenvectors often provide insight into the behavior of the transformation that cannot be obtained from real eigenvalues and eigenvectors.

5. What is the difference between eigenvalues and diagonalization?

Eigenvalues and diagonalization are related concepts, but they are not the same. Eigenvalues refer to the scalar values by which an eigenvector is scaled when transformed by a linear transformation. Diagonalization refers to the process of finding a diagonal matrix that is similar to the original matrix, which means that they have the same eigenvalues. In other words, diagonalization is finding a different basis for the same linear transformation, where the new basis consists of eigenvectors.

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