Eigenvalue & Eigenvector Calculator

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What this eigenvalue & eigenvector calculator does

This calculator computes the eigenvalues and eigenvectors of a 2×2 real matrix. You enter the four entries of your matrix, and the tool returns:

Both eigenvalues $λ_{1}$ and $λ_{2}$
At least one eigenvector for each eigenvalue (often normalized)
Intermediate quantities such as the trace and determinant, when relevant

It is designed for students, engineers, and data scientists who want to understand how a linear transformation stretches, compresses, or flips the plane along special directions.

Definition of eigenvalues and eigenvectors

Let $A$ be a square matrix. An eigenvector of $A$ is a nonzero vector $v$ such that

$A v = λ v,$

for some scalar $λ$ . The scalar $λ$ is called the eigenvalue associated with $v$ . Intuitively, applying the linear transformation represented by $A$ to $v$ does not change its direction, only its length (and possibly its sign).

For a 2×2 matrix, you can think of $A$ as a transformation of the plane. Most vectors in the plane will be rotated, stretched, or sheared in complicated ways, but eigenvectors mark the directions that are preserved by the transformation up to a scale factor.

Characteristic polynomial for a 2×2 matrix

Consider a 2×2 matrix

$A = (\begin{matrix} a & b \\ c & d \end{matrix}) .$

The eigenvalues are obtained by solving the characteristic equation

$det (A - λ I) = 0$

where $I$ is the 2×2 identity matrix. For this matrix, we have

$A - λ I = (\begin{matrix} a - λ & b \\ c & d - λ \end{matrix})$

$| A - λ I | = (a - λ) (d - λ) - b c .$

Expanding this determinant yields a quadratic polynomial in $λ$ :

$\lambda^2 - (a + d) \lambda + (a d - b c) = 0$

It is often convenient to express this in terms of the trace and determinant of $A$ :

$tr (A) = a + d$
$\det (A) = a d - b c$

Then the characteristic polynomial becomes

$λ^{2} - (tr (A)) λ + | A | = 0$

The two roots of this quadratic are exactly the eigenvalues $λ_{1}$ and $λ_{2}$ . When the discriminant

$Δ = {(tr (A))}^{2} - 4 \det (A)$

is positive, both eigenvalues are real and distinct. If $Δ = 0$ , there is a repeated real eigenvalue. If $Δ < 0$ , then the eigenvalues form a complex conjugate pair.

How eigenvectors are computed

Once an eigenvalue $λ$ has been found, the associated eigenvectors are obtained by solving the homogeneous system

$(A - λ I) v = 0 .$

For our 2×2 matrix $A$ , this system is

$(\begin{matrix} a - λ & b \\ c & d - λ \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix}) .$

This corresponds to a pair of linear equations in the unknowns $x$ and $y$ . Because $λ$ is an eigenvalue, these equations are not independent; they describe a line of solutions through the origin. Any nonzero vector on this line is an eigenvector associated with $λ$ .

In practice, we often set one component (for example, $x$ ) to a convenient value such as 1, then solve for the other component $y$ . Finally, we may normalize the resulting vector so that it has length 1:

$\hat{v} = \frac{1}{∥ v ∥} v .$

Normalization does not change the direction of the eigenvector, but it makes it easier to compare eigenvectors and can be numerically more stable in larger computations.

Worked example: eigenvalues and eigenvectors of a 2×2 matrix

Consider the matrix

$A = (\begin{matrix} 3 & 1 \\ 4 & 2 \end{matrix}) .$

If you are using the calculator, you would enter:

$a_{11} = 3$
$a_{12} = 1$
$a_{21} = 4$
$a_{22} = 2$

Step 1: trace and determinant

Compute the trace and determinant:

$tr (A) = 3 + 2 = 5$
$det (A) = 3 \cdot 2 - 1 \cdot 4 = 6 - 4 = 2$

Step 2: characteristic polynomial and eigenvalues

The characteristic equation is

$λ^{2} - 5 λ + 2 = 0 .$

Using the quadratic formula,

$λ = \frac{5 \pm \sqrt{5^{2} - 4 \cdot 2}}{2} = \frac{5 \pm \sqrt{25 - 8}}{2} = \frac{5 \pm \sqrt{17}}{2} .$

So the two (real) eigenvalues are

$λ_{1} = \frac{5 + \sqrt{17}}{2}, λ_{2} = \frac{5 - \sqrt{17}}{2} .$

(If the discriminant had been negative, the calculator would report complex eigenvalues in a suitable form.)

Step 3: eigenvector for $λ_{1}$

To find an eigenvector for $λ_{1}$ , solve

$(A - λ_1 I) v = 0 .$

This gives

$(\begin{matrix} 3 - λ_{1} & 1 \\ 4 & 2 - λ_{1} \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix}) .$

From the first row, we have

$(3 - λ_{1}) x + y = 0$

$y = (λ_{1} - 3) x .$

Setting $x = 1$ yields an eigenvector

$v 1 = (\begin{matrix} 1 \\ λ_{1} - 3 \end{matrix})$

Any nonzero scalar multiple of this vector is also an eigenvector associated with $λ_{1}$ . The calculator may optionally normalize this vector before displaying it.

Step 4: eigenvector for $λ_{2}$

Repeating the same process for $λ_{2}$ , we solve

$(A - λ_{2} I) v = 0,$

leading to

$(3 - λ_{2}) x + y = 0 .$

Again setting $x = 1$ gives

$v 2 = (\begin{matrix} 1 \\ λ_{2} - 3 \end{matrix})$

as an eigenvector associated with $λ_{2}$ .

Interpreting this example

The eigenvalues $λ_{1}$ and $λ_{2}$ describe how vectors along the special directions $v_{1}$ and $v_{2}$ are scaled when multiplied by $A$ . One direction is stretched by a factor of $λ_{1}$ , and the other by $λ_{2}$ . If one eigenvalue is greater than 1 in magnitude and the other is less than 1, the transformation has both expanding and contracting directions.

How to use the calculator

Identify your 2×2 matrix $A$ and write it in the form $A = (\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}) .$
Enter each entry of $A$ into the corresponding input field.
Click the button to compute eigenvalues and eigenvectors.
Review the reported eigenvalues $λ λ_{1}, λ_{2}$ , the associated eigenvectors, and any displayed intermediate quantities such as the trace and determinant.

If the matrix has complex eigenvalues, the calculator will display them in a suitable numeric form. For real eigenvalues, you will typically see decimal approximations, and possibly exact symbolic expressions for simple cases.

Interpreting the results

When you look at the output of the calculator, focus on three aspects:

Sign and magnitude of eigenvalues: A positive eigenvalue greater than 1 represents stretching along its eigenvector; a value between 0 and 1 represents compression; a negative eigenvalue represents stretching combined with a flip in direction.
Relative sizes of eigenvalues: In iterative processes or dynamical systems, the eigenvalue with the largest magnitude often dominates long-term behavior.
Orientation of eigenvectors: Eigenvectors tell you the special directions in the plane that remain invariant up to scale. They can correspond to principal directions of deformation, principal components in data, or natural modes in vibrations.

Comparison with other eigenvalue tools

The table below contrasts this focused 2×2 calculator with more general matrix solvers you might encounter in software packages or computer algebra systems.

Tool	Matrix size	Focus	Typical user
This 2×2 eigenvalue & eigenvector calculator	Fixed at 2×2	Fast computations with clear explanation and educational context	Students, instructors, and practitioners needing quick insight
General numerical linear algebra libraries	Arbitrary sizes (often large)	High-performance computation, minimal exposition	Engineers, scientists, data engineers running large models
Computer algebra systems	Symbolic and numeric matrices	Exact symbolic eigenvalues and eigenvectors	Researchers and advanced users needing symbolic results

Assumptions and limitations

Matrix size: The calculator is limited to 2×2 matrices. Larger matrices require more general tools.
Inputs: It expects numeric entries (typically real numbers). Non-numeric input cannot be processed.
Complex eigenvalues: If the discriminant is negative, the eigenvalues are complex; these may be displayed in approximate numeric form. Detailed complex eigenvector calculations may be limited.
Rounding: Results may be rounded to a reasonable number of decimal places. Small numerical differences can occur due to floating-point arithmetic.
Use case: The tool is intended for learning, checking homework, and preliminary analysis. It should not be used as the sole basis for safety-critical or certified engineering design.

Common questions

Can a 2×2 matrix have complex eigenvalues?

Yes. When the discriminant $Δ = (tr (A)^{2}) - 4 det (A)$ is negative, the two eigenvalues form a complex conjugate pair. In that case, the calculator reports complex values rather than real numbers.

Are eigenvectors unique?

No. If $v$ is an eigenvector, then any nonzero scalar multiple $c v$ with $c \neq 0$ is also an eigenvector for the same eigenvalue. Typically, a normalized eigenvector is shown for convenience, but many equivalent choices exist.

What happens if the matrix has a repeated eigenvalue?

If the characteristic polynomial has a double root, the matrix has a repeated eigenvalue. There may still be two independent eigenvectors (the matrix is diagonalizable), or only one independent eigenvector (the matrix is defective). A 2×2 defective matrix will yield only a one-dimensional eigenspace for that eigenvalue.

How is this related to stability analysis?

In simple linear dynamical systems, eigenvalues help determine whether solutions grow, decay, or oscillate over time. For example, an eigenvalue with magnitude less than 1 (in discrete time) or negative real part (in continuous time) often corresponds to stable behavior along that eigenvector direction.

Eigenvalue & Eigenvector Calculator

What this eigenvalue & eigenvector calculator does

Definition of eigenvalues and eigenvectors

Characteristic polynomial for a 2×2 matrix

How eigenvectors are computed

Worked example: eigenvalues and eigenvectors of a 2×2 matrix

Step 1: trace and determinant

Step 2: characteristic polynomial and eigenvalues

Step 3: eigenvector for λ1

Step 4: eigenvector for λ2

Interpreting this example

How to use the calculator

Interpreting the results

Comparison with other eigenvalue tools

Assumptions and limitations

Common questions

Can a 2×2 matrix have complex eigenvalues?

Are eigenvectors unique?

What happens if the matrix has a repeated eigenvalue?

How is this related to stability analysis?

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Step 3: eigenvector for $λ_{1}$

Step 4: eigenvector for $λ_{2}$