Free Eigenvalue and Eigenvector Calculator

Matrix Eigenvalues and Eigenvectors Explained

Working with square matrices often requires determining eigenvalues and eigenvectors—a foundational step in matrix diagonalization, stability analysis, and dimensionality reduction. This eigenvalue and eigenvector calculator handles 2×2 and 3×3 matrices, automatically computing both the eigenvalues and their associated eigenvectors. It also reveals the trace and determinant of the matrix, giving you a complete picture of its algebraic properties. Whether you are a student learning linear algebra or a professional verifying results, this tool simplifies the entire process.

Working with a 2×2 Matrix

A generic 2×2 matrix A has four entries:

$$A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$$

When you enter these numbers into the calculator, it uses exactly this layout—so be careful to place each element in its correct position.

Trace and Determinant

Before solving for eigenvalues, it's helpful to compute two key numbers:

• Trace (tr(A)): the sum of the diagonal entries, which also equals the sum of the eigenvalues (counting multiplicities). For a 2×2 matrix:

$$\text{tr}(A) = a_{11} + a_{22}$$

• Determinant (det(A)): a scalar that encodes essential information about the matrix, such as whether it is invertible. For a 2×2 matrix:

$$\det(A) = a_{11}a_{22} - a_{12}a_{21}$$

The calculator displays these values automatically if you expand the corresponding section, and you can use the formulas above to double‑check the results.

Finding Eigenvalues

A 2×2 matrix has two eigenvalues, $\lambda_1$ and $\lambda_2$. They satisfy the equation:

$$A \mathbf{v} = \lambda \mathbf{v}$$

where $\mathbf{v}$ is a non‑zero column vector called an eigenvector. An equivalent form is:

$$(A - \lambda I) \mathbf{v} = 0$$

with $I$ being the 2×2 identity matrix. Using the trace and determinant, the eigenvalues can be obtained directly:

$$\lambda_{1,2} = \frac{\text{tr}(A) \pm \sqrt{(\text{tr}(A))^2 - 4\det(A)}}{2}$$

Some matrices have a single eigenvalue (a repeated root). For example, matrices of the form:

$$\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}$$

have only one eigenvalue, $\lambda = a$. You can examine such cases with the calculator to see how they behave.

Finding Eigenvectors

To obtain an eigenvector for a given eigenvalue $\lambda$, solve:

$$(A - \lambda I) \mathbf{v} = 0$$

If the matrix has only one eigenvalue, it may still have multiple independent eigenvectors. Take the identity matrix:

$$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$

It has a double eigenvalue $\lambda = 1$ and two eigenvectors: $\mathbf{v}_1 = (1,0)^T$ and $\mathbf{v}_2 = (0,1)^T$. Also remember that if $\mathbf{v}$ is an eigenvector, any non‑zero scalar multiple of it is also an eigenvector for the same eigenvalue.

Extending to 3×3 Matrices

A 3×3 matrix has the form:

$$A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$$

Definitions of trace and eigenvectors remain similar, but the determinant becomes more complex:

$$\begin{aligned} \det(A) &= a_{11}(a_{22}a_{33} - a_{23}a_{32}) \\ &\quad - a_{12}(a_{21}a_{33} - a_{23}a_{31}) \\ &\quad + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \end{aligned}$$

The eigenvalue condition $A\mathbf{v} = \lambda \mathbf{v}$ still holds, but the characteristic polynomial is now cubic:

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

For 2×2 matrices, this leads to a quadratic equation; for 3×3 matrices, it becomes a cubic equation, which is less straightforward to solve by hand. The calculator handles this automatically, performing the necessary polynomial root‑finding to deliver the eigenvalues and eigenvectors for any real‑valued 3×3 matrix.

Complex Eigenvalues

Quadratic and cubic equations sometimes have no real solutions—that is, no real number satisfies the characteristic polynomial. In such cases, eigenvalues are complex numbers, each having a real part and an imaginary part. The imaginary unit $i$ satisfies $i^2 = -1$. While real‑world problems often restrict attention to real eigenvalues, many applications (e.g., vibration analysis, control theory) rely on complex eigenvalues and eigenvectors. The calculator computes both real and complex results, displaying them together so you can pick what you need. If only real eigenvalues matter, simply ignore those containing an $i$.

With this tool, you can explore the eigenvalues and eigenvectors of any 2×2 or 3×3 square matrix, gain insight into its structure, and verify your own manual calculations—all in a few clicks.