Hi, well I have the next question:
How can I show that the sum of two eigen values is not another eigen value?
I know that I can give an example, but is ther another way to solve?, please help...
To show that the sum of two eigenvalues is not necessarily another eigenvalue, you can use the definition of eigenvectors and eigenvalues.
Let A be a square matrix, and λ₁ and λ₂ be two eigenvalues of A. Also, let v₁ and v₂ be the corresponding eigenvectors.
According to the definition of eigenvalues and eigenvectors, we have:
Av₁ = λ₁v₁ (Equation 1)
Av₂ = λ₂v₂ (Equation 2)
Now, let's consider the sum of the two eigenvectors: v = v₁ + v₂.
To determine if v is an eigenvector, we need to check if there exists a scalar λ such that Av = λv.
Multiply equation 1 by λ₂ and equation 2 by λ₁:
λ₂Av₁ = λ₁λ₂v₁ (Equation 3)
λ₁Av₂ = λ₁λ₂v₂ (Equation 4)
Add equations 3 and 4:
λ₂Av₁ + λ₁Av₂ = λ₁λ₂v₁ + λ₁λ₂v₂
= λ₁λ₂(v₁ + v₂) (Equation 5)
Now, let's calculate Av:
A(v₁ + v₂) = Av₁ + Av₂ (distributive property of matrix multiplication)
= λ₁v₁ + λ₂v₂ (substituting equations 1 and 2)
= λ₁v₁ + λ₂v₂ + 0v (adding 0v to ensure both sides have the same dimensions)
= λ₁v₁ + λ₂v₂ + 0(v₁ + v₂)
= λ₁v₁ + λ₂v₂ + 0v
Comparing this with equation 5, we see that λ₁λ₂ is the scalar λ we are looking for. However, it is not necessarily equal to the sum of the original eigenvalues (λ₁ + λ₂).
Therefore, the sum of two eigenvalues is not necessarily another eigenvalue.