Symmetric matrix real eigenvalues

In summary, to prove that a symmetric (2x2) matrix always has real eigenvalues, you can use the quadratic formula. By expanding the determinant and using the quadratic formula, you can show that the term under the square root will always be positive by writing it as a sum of squares. The -4ad term can be simplified to (a+d)^2 - 4ad, which is also always positive.
  • #1
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Homework Statement



Prove a symmetric (2x2) matrix always has real eigenvalues. The problem shows the matrix as {(a,b),(b,d)}.


Homework Equations



The problem says to use the quadratic formula.

The Attempt at a Solution



From the determinant I get (a-l)(d-l) - b^2 = 0 which expands to l^2 - (a+d)l + (ad - b^2) = 0

Using the quadratic formula I get for under the square root: (a + d)^2 - 4(ad-b^2)
How can I show that this is always positive?

Thanks for the help
 
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  • #2
You can write the square root term as a sum of squares, which is always positive.
 
  • #3
How do you deal with the -4ad term? I tried to factor it but couldn't figure out how
 
  • #4
Can you see what (a+d)^2 - 4ad is?
 

1. What is a symmetric matrix?

A symmetric matrix is a square matrix where the elements are symmetric with respect to the main diagonal. This means that the element at row i and column j is equal to the element at row j and column i.

2. What are real eigenvalues?

Real eigenvalues are the set of values that, when multiplied by a given matrix, yield a scalar multiple of the original matrix. In other words, they are the values that satisfy the equation Ax = λx, where A is the given matrix, x is a non-zero vector, and λ is the eigenvalue.

3. Why are real eigenvalues important in symmetric matrices?

In symmetric matrices, real eigenvalues have special properties that make them useful in many applications. For example, symmetric matrices with real eigenvalues are easier to diagonalize, and their eigenvalues can provide information about the matrix's behavior and properties.

4. How can I find the real eigenvalues of a symmetric matrix?

The real eigenvalues of a symmetric matrix can be found by solving the characteristic equation det(A-λI) = 0, where A is the given matrix, λ is the eigenvalue, and I is the identity matrix. This equation will result in a polynomial of degree n, where n is the size of the matrix, and the roots of this polynomial will be the real eigenvalues.

5. Can a symmetric matrix have complex eigenvalues?

No, a symmetric matrix can only have real eigenvalues. This is because the eigenvalues of a symmetric matrix are the roots of a real polynomial, and complex roots always occur in conjugate pairs. Since a symmetric matrix has real coefficients, its eigenvalues must also be real.

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