Section 46.1 Terminology and notation
Each of the definitions below are for a linear operator \(\funcdef{T}{V}{V}\) on a finite-dimensional vector space \(V\text{.}\)
- determinant
the determinant of the matrix \(\matrixOf{T}{B}\) for any choice of domain space basis \(\basisfont{B}\)
- trace
the trace of the matrix \(\matrixOf{T}{B}\) for any choice of domain space basis \(\basisfont{B}\)
- eigenvector
a nonzero vector \(\uvec{x}\) in the domain space such that the image vector \(T(\uvec{x})\) is a scalar multiple of \(\uvec{x}\)
- eigenvalue
a scalar for which there exists an eigenvector \(\uvec{x}\) of operator \(T\) with \(T(\uvec{x}) = \lambda \uvec{x}\)
- eigenspace
the subspace of the domain space consisting of all eigenvectors of \(T\) that correspond to a specific eigenvalue \(\lambda\text{,}\) along with the zero vector
- \(E_\lambda(T)\)
notation for the eigenspace of operator \(T\) corresponding to the eigenvalue \(\lambda\)
- characteristic polynomial
the degree-\(n\) polynomial in the variable \(\lambda\) obtained by computing \(\det(\lambda I - T)\)
- \(c_T(\lambda)\)
notation for the characteristic polynomial of operator \(T\)
- characteristic equation
the polynomial equation \(\det(\lambda I - T) = 0\)