1) Definition
• Let
L:V → V1
be the linear transformation of n-dimensional vector space into itself (a linear operator on V)
Then, λ is an eigenvalue of L if there exists a non-zero vector x in V such that
L(x) = λx
In other words, λ is a scalar associated with vector x to represent a linear transformation.
• If λ = eigenvalue, thenx = eigenvector (an eigenvector is always associated with an eigenvalue)
Eg: If L(x) = 5x , 5 is the eigenvalue andx is the eigenvector.
* λ can be either real or complex, as will be shown later.
An example about the concept of eigenvalue and eigenvector, based on a linear transformation:
(Read the matrix [a , b] as entry [a] in the first row and entry [b] in the second row)
• Given that L:R2 → R2 be linear operator defined by
L([a1 , a2]) = [-a2 , a1]
To find the eigenvalues of L and the associated eigenvectors, we have to find λ such that
L([a1 , a2]) = λ[a1, a2]
Since [-a2 , a1] = λ[a1 , a2], equate them together, and we can find that
λa1 = -a2 (1) and λa2 = a1 (2)
By substituting (2) into (1), we obtain
λ(λa2) = -a2
λ2a2 = - a2
λ2 = -1
Since the square root of -1 is equal to the complex numberi, we can conclude that λ = ±i
Because the eigenvalues are not real, we can say that there is no vector [a1 , a2] in R2 such that L([a1 , a2]) is parallel to [a1 , a2].
But if L is mapped from C2 into C2 , then L has eigenvalue i (eigenvector [i , 1]) and eigenvalue -i (eigenvector [-i , 1]
