Evaluating B4: Finding the Eigenvalues & Eigenvectors

In summary, the problem is asking to evaluate B^4 for a matrix B with characteristic polynomial λ^2-λ√6+3. Using Cayley-Hamilton's theorem, it can be simplified to a much simpler expression involving B^2, which can then be solved using basic matrix operations.
  • #1
shaon0
48
0

Homework Statement


Let B be a matrix with characteristic polynomial λ2-λ√6+3. Evaluate B4.

Homework Equations


Bn=PDnP-1

The Attempt at a Solution


I can find the eigenvalues from the characteristic equation and those would form the diagonal entries of D. But how would I find P, which contains the eigenvectors, if I don't have a matrix?

Side Note: How can I quickly find an inverse for a 2 by 2 matrix. Is it just dividing the 2x2 matrix by it's determinant, then negating the diagonal entries going from a11 to a22 and swapping a12 and a21?
 
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  • #2
Hi shaon0! :smile:

shaon0 said:

Homework Statement


Let B be a matrix with characteristic polynomial λ2-λ√6+3. Evaluate B4.

Homework Equations


Bn=PDnP-1

The Attempt at a Solution


I can find the eigenvalues from the characteristic equation and those would form the diagonal entries of D. But how would I find P, which contains the eigenvectors, if I don't have a matrix?

Side Note: How can I quickly find an inverse for a 2 by 2 matrix. Is it just dividing the 2x2 matrix by it's determinant, then negating the diagonal entries going from a11 to a22 and swapping a12 and a21?

Let's first see how far we can get before concluding you have too little information shall we?

What did you find for the eigenvalues?
What do you get if you calculate D4?


On your side note: almost, but you need to swap the diagonal entries a11 and a22, and negate a12 and a21.
Check by multiplying your matrix with the supposed inverse. It should yield the identity matrix.
 
  • #3
I like Serena said:
Hi shaon0! :smile:
Let's first see how far we can get before concluding you have too little information shall we?

What did you find for the eigenvalues?
What do you get if you calculate D4?On your side note: almost, but you need to swap the diagonal entries a11 and a22, and negate a12 and a21.
Check by multiplying your matrix with the supposed inverse. It should yield the identity matrix.

Hi I like Serena :);

I've found the answer. Thanks for the help, needed another to look at the problem.
 
Last edited:
  • #4
You don't really need to calculate the eigenvalues or [itex]D^4[/itex] for this.

Cayley-Hamilton says that

[tex]B^2=\sqrt{6}B-3I[/tex]

Thus

[tex]B^4=B^2*B^2=B^2(\sqrt{6}B-3I)=\sqrt{6}B^3-3B^2=\sqrt{6}B(\sqrt{6}B-3I)-3(\sqrt{6}B-3I)=...[/tex]
 
  • #5
Neat! :smile:

Btw, it's also neat to see how Bn=PDnP-1 cancels out. :cool:
 

1. What is the purpose of evaluating B4 and finding the eigenvalues and eigenvectors?

Evaluating B4 and finding the eigenvalues and eigenvectors allows us to understand the behavior and properties of a matrix. This information is useful in various applications, such as solving systems of linear equations, computing powers of a matrix, and understanding transformations in linear algebra.

2. How do you calculate the eigenvalues and eigenvectors of a matrix?

To find the eigenvalues and eigenvectors of a matrix, we first need to calculate the characteristic polynomial of the matrix. Then, we solve for the roots of the characteristic polynomial, which are the eigenvalues. Finally, we use the eigenvalues to find the corresponding eigenvectors by solving a system of linear equations.

3. What is the significance of the eigenvalues and eigenvectors?

Eigenvalues and eigenvectors have many important applications in mathematics and science. They can be used to solve differential equations, analyze financial data, and understand the behavior of dynamical systems. In addition, they are essential for diagonalization and diagonalization is crucial for many matrix computations.

4. Can a matrix have complex eigenvalues and eigenvectors?

Yes, a matrix can have complex eigenvalues and eigenvectors. This occurs when the characteristic polynomial has complex roots. In this case, the eigenvalues and eigenvectors will also be complex numbers. However, in many real-world applications, the eigenvalues and eigenvectors will be real numbers.

5. How do you determine the number of eigenvalues and eigenvectors of a matrix?

The number of eigenvalues and eigenvectors of a matrix is equal to the size of the matrix. For example, a 3x3 matrix will have three eigenvalues and three corresponding eigenvectors. However, some eigenvalues may be repeated, and some matrices may have fewer than the expected number of eigenvectors. This can occur when the matrix is defective, meaning it cannot be diagonalized.

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