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\)
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
\(\det(\lambda I - T) = 0\)
