Eigenvalue

Definition

The eigenvectors of a square matrix are the nonzero vectors that, after being multiplied by the matrix, either remain proportional to the original vector (i.e., change only in magnitude, not in direction) or become zero. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector changes when multiplied by the matrix. The prefix eigen- is adopted from the German word “eigen” for “own” in the sense of a characteristic description. The eigenvectors are sometimes also called characteristic vectors. Similarly, the eigenvalues are also known as characteristic values.

The mathematical expression of this idea is as follows: if A is a square matrix, a nonzero vector $\upsilon$ is an eigenvector of $A$ if there is a scalar $\lambda$ (lambda) such that:

$$ A\upsilon = \lambda \upsilon $$

The scalar $\lambda$ is said to be the eigenvalue of $A$ corresponding to $\upsilon$. An eigenspace of $A$is the set of all eigenvectors...