Characteristic Polynomial for A with a=-1

In summary, the conversation discusses the use of a field, denoted as L, and a matrix A with elements from L. The matrix has a characteristic polynomial that is given by x^4 - (a+1)x^3 + (a-1)x^2 + (a+1)x - a. The question is to show that this polynomial is correct for a=1 in any field. However, when calculating for a=-1, a different polynomial is obtained. It is suggested that the characteristic polynomial may not be accurate for a=-1.
  • #1
keddelove
3
0
And again a question:

L is a field for which [tex] a \in L [/tex]. The matrix

[tex]
A = \frac{1}{2}\left( {\begin{array}{*{20}c}
1 & 1 & 1 & 1 \\
1 & a & { - 1} & { - a} \\
1 & { - 1} & 1 & { - 1} \\
1 & { - a} & { - 1} & a \\
\end{array}} \right)
[/tex]

has the characteristic polynomial

[tex]
x^4 - \left( {a + 1} \right)x^3 + \left( {a - 1} \right)x^2 + \left( {a + 1} \right)x - a
[/tex]

I need to show that this information is correct for a=1 in any field.

My problem is that when i calculate the polynomial for a=-1 I end up with another characteristic polynomial than the one given. Maybe I'm going about it the wrong way. Suggestions or pointers are very welcome
 
Physics news on Phys.org
  • #2
I don't understand the question...

And if you want to show something is correct for a=1, then why are you looking at a=-1?
 
  • #3
i conjecture he meant the characteristic polynomial is accurate for a any element of any field, and yet it failed for a = -1.
 
  • #4
Oops, should have stated:

Show that this is correct for a=-1 in any field.
 

What is a characteristic polynomial?

A characteristic polynomial is a polynomial function that is associated with a square matrix. It is used to find the eigenvalues of the matrix, which are important values that describe the behavior of the matrix.

Why is the characteristic polynomial important?

The characteristic polynomial is important because it helps us understand the structure and properties of a matrix. It allows us to find the eigenvalues of the matrix, which are useful in solving various problems in mathematics and science.

How do you find the characteristic polynomial of a matrix?

To find the characteristic polynomial of a matrix, one must first find the determinant of the matrix. Then, the characteristic polynomial is formed by setting the determinant equal to a variable (usually denoted as lambda) and solving for lambda. This process is known as finding the characteristic equation.

What is the degree of a characteristic polynomial?

The degree of a characteristic polynomial is equal to the size of the matrix. For example, a 3x3 matrix will have a characteristic polynomial of degree 3. This is because the degree of the polynomial is determined by the highest power of lambda in the characteristic equation.

Can the characteristic polynomial have complex roots?

Yes, the characteristic polynomial can have complex roots. This is because matrices can have complex eigenvalues, and the characteristic polynomial is used to find these eigenvalues. In fact, for a 2x2 matrix, the characteristic polynomial will always have complex roots if the matrix has non-real eigenvalues.

Similar threads

  • Linear and Abstract Algebra
Replies
5
Views
821
  • Linear and Abstract Algebra
Replies
2
Views
882
  • Linear and Abstract Algebra
Replies
2
Views
918
Replies
6
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
2K
  • Linear and Abstract Algebra
Replies
8
Views
995
  • Linear and Abstract Algebra
Replies
19
Views
2K
Replies
27
Views
1K
  • Linear and Abstract Algebra
Replies
14
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
2K
Back
Top