Section 24.2 Terminology and notation
The following definitions are relative to a given \(n \times n\) matrix \(A\text{.}\)
- eigenvector
a nonzero vector \(\uvec{x}\) in \(\R^n\) such that \(A\uvec{x}\) is a scalar multiple of \(\uvec{x}\)
- eigenvalue
a scalar for which there exists an eigenvector \(\uvec{x}\) of \(A\) with \(A\uvec{x}=\lambda\uvec{x}\)
Note 24.2.1.
Eigenvectors and eigenvalues go together in pairs, with the connection between the two provided by the equality \(A\uvec{x}=\lambda\uvec{x}\text{.}\) In this situation, we say that the two objects correspond to each other. So we might say that \(\uvec{x}\) is an eigenvector of \(A\) that corresponds to the eigenvalue \(\lambda\text{.}\) Equivalently, we might say that \(\lambda\) is an eigenvalue of \(A\) that corresponds to the eigenvector \(\uvec{x}\text{.}\) However, note that an eigenvalue can correspond to many eigenvectors (in fact, an infinite number of them), an eigenvector must correspond to exactly one eigenvalue.
- eigenspace
the subspace of \(\R^n\) consisting of all eigenvectors of \(A\) that correspond to a specific eigenvalue \(\lambda\text{,}\) along with the zero vector
- \(E_\lambda(A)\)
notation for the eigenspace of matrix \(A\) corresponding to the eigenvalue \(\lambda\)
Note 24.2.2.
In other resources you may seem the terms characteristic vector, characteristic value, and characteristic space used in place of the terminology introduced above.
- characteristic polynomial
the degree-\(n\) polynomial in the variable \(\lambda\) obtained by computing \(\det(\lambda I - A)\)
- \(c_A(\lambda)\)
notation for the characteristic polynomial of matrix \(A\)
- characteristic equation
the polynomial equation \(\det(\lambda I - A) = 0\)