stylistic:tone, structural:reformatting, information:filtering, linguistic:paraphrasing
Left and right eigenvectors

Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word "eigenvector" in the context of matrices almost always refers to a right eigenvector, namely a column vector that right multiplies the n × n {\displaystyle n\times n} matrix A {\displaystyle A} in the defining equation, equation (1), A v = λ v. {\displaystyle A\mathbf {v} =\lambda \mathbf {v}.}

The eigenvalue and eigenvector problem can also be defined for row vectors that left multiply matrix A {\displaystyle A}. In this formulation, the defining equation is u A = κ u, {\displaystyle \mathbf {u} A=\kappa \mathbf {u},}

where κ {\displaystyle \kappa } is a scalar and u {\displaystyle u} is a 1 × n {\displaystyle 1\times n} matrix. Any row vector u {\displaystyle u} satisfying this equation is called a left eigenvector of A {\displaystyle A} and κ {\displaystyle \kappa } is its associated eigenvalue. Taking the transpose of this equation, A T u T = κ u T. {\displaystyle A^{\textsf {T}}\mathbf {u} ^{\textsf {T}}=\kappa \mathbf {u} ^{\textsf {T}}.}

Comparing this equation to equation (1), it follows immediately that a left eigenvector of A {\displaystyle A} is the same as the transpose of a right eigenvector of A T {\displaystyle A^{\textsf {T}}}, with the same eigenvalue. Furthermore, since the characteristic polynomial of A T {\displaystyle A^{\textsf {T}}} is the same as the characteristic polynomial of A {\displaystyle A}, the left and right eigenvectors of A {\displaystyle A} are associated with the same eigenvalues.