What Are Left and Right Eigenvalues?

In standard linear algebra, “left” and “right” usually distinguish eigenvectors, not two different sets of eigenvalues. A right eigenvector is multiplied by a matrix on its left; a left eigenvector is a row vector multiplied by the matrix on its right. For an ordinary real or complex square matrix, both kinds correspond to the same eigenvalues.

The distinction matters when you calculate eigenvectors or use them in applications. Here are the definitions, a worked example, and the transpose convention to watch for with complex matrices.

Right and left eigenvector definitions

Let $A$ be an $n \times n$ matrix. A nonzero column vector $v$ is a right eigenvector for eigenvalue $\lambda$ if

$$Av = \lambda v.$$

A nonzero row vector $w^T$ is a left eigenvector for $\lambda$ if

$$w^T A = \lambda w^T.$$

The superscript $T$ means transpose. Equivalently, if you write the left eigenvector as a column $u$, it satisfies $A^T u = \lambda u$. Thus, right eigenvectors of $A$ are found by solving a system involving $A - \lambda I$, while left eigenvectors can be found by solving one involving $A^T - \lambda I$.

The scalar $\lambda$ is an eigenvalue in either equation. For a square matrix over a field such as $\mathbb{R}$ or $\mathbb{C}$, the eigenvalues are found by setting the characteristic polynomial to zero:

$$\det(A - \lambda I) = 0.$$

Transposing does not change a determinant, so $A$ and $A^T$ have the same characteristic polynomial and the same eigenvalues, including their algebraic multiplicities.

Worked example

Consider

$$A = \begin{pmatrix}1 & 1 \\ 0 & 2\end{pmatrix}.$$

Its characteristic polynomial is

$$\det(A - \lambda I) = \det\begin{pmatrix}1-\lambda & 1 \\ 0 & 2-\lambda\end{pmatrix} = (1-\lambda)(2-\lambda),$$

so the eigenvalues are $1$ and $2$.

For $\lambda = 1$, a right eigenvector solves $(A-I)v=0$. This gives $v_2=0$, so one choice is $v=(1,0)^T$. A left eigenvector solves $w^T(A-I)=0$. Writing $w^T=(a,b)$ gives $a+b=0$, so one choice is $w^T=(1,-1)$.

For $\lambda = 2$, solving $(A-2I)v=0$ gives $v_1=v_2$. One right eigenvector is $v=(1,1)^T$. Solving $w^T(A-2I)=0$ gives $a=0$, so one left eigenvector is $w^T=(0,1)$.

The left and right eigenvectors are not generally the same. In this example, the eigenvalue $1$ has right eigenvector $(1,0)^T$ and left eigenvector $(1,-1)$, while the eigenvalue $2$ has right eigenvector $(1,1)^T$ and left eigenvector $(0,1)$.

How to calculate them

For each eigenvalue $\lambda$:

1. Find right eigenvectors by solving $(A-\lambda I)v=0$.
2. Find left eigenvectors by solving $w^T(A-\lambda I)=0$, or equivalently $(A^T-\lambda I)w=0$ for a column $w$.
3. Choose any nonzero solution; all solutions, including zero, form the corresponding eigenspace.

For a square matrix, the left and right eigenspaces for a given eigenvalue have the same dimension. This follows because a matrix and its transpose have the same rank, and both systems are square. Their vectors need not be equal or proportional.

Complex matrices and the conjugate transpose

For complex matrices, the algebraic definition above uses the ordinary transpose: $w^T A = \lambda w^T$. Some authors instead express a left eigenvector using the conjugate transpose, writing $u^*A = \lambda u^*$, where $*$ means transpose and complex conjugation. Under that convention, the associated column equation is $A^*u = \overline{\lambda}u$. The conjugate appears because conjugating the row equation also conjugates its scalar. Check which convention is being used before comparing formulas.

Common mistakes

• Assuming the eigenvectors must match: Left and right eigenvectors usually differ, even though they correspond to the same eigenvalue.
• Using the wrong matrix: To find left eigenvectors as columns, solve with $A^T$, not $A$.
• Conjugating without checking the definition: The ordinary left-eigenvector equation uses transpose; a conjugate-transpose convention changes the displayed eigenvalue relation.
• Applying the claim outside ordinary matrix algebra: The same-eigenvalues result uses standard square matrices over a commutative field. In settings such as quaternion matrices, definitions and results can differ.