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Eigenvector Calculator

Use this free online eigenvector calculator to find the eigenvector for a given eigenvalue of a 2x2 matrix. Enter the matrix and eigenvalue to solve (A - lambda I)v = 0 with step-by-step workings.

Enter Values

Top-left entry of the matrix

Top-right entry of the matrix

Bottom-left entry of the matrix

Bottom-right entry of the matrix

The eigenvalue to find the eigenvector for. Use the eigenvalue calculator first if needed.

Result

Enter values above and click Calculate to see your result.

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Formula

Solve (A - lambda I)v = 0 for nonzero v

Subtract the eigenvalue times the identity matrix from A, then solve the resulting homogeneous system. Any nonzero solution is an eigenvector. The eigenvector direction is unique (up to scalar multiples).

Worked Example

A = [[4, 1], [2, 3]], eigenvalue = 5 Step 1: A - 5I = [[4-5, 1], [2, 3-5]] = [[-1, 1], [2, -2]] Step 2: Row reduce: Row 1 gives -v1 + v2 = 0, so v2 = v1 Step 3: Choose v1 = 1, then v2 = 1 Result: Eigenvector = [1, 1] (or any scalar multiple like [2, 2], [-1, -1]).

What Is an Eigenvector?

An eigenvector is a nonzero vector that does not change direction when a linear transformation (matrix) is applied to it. It only gets scaled by a factor (the eigenvalue). Finding eigenvectors is essential for understanding how matrices act on space and for decomposing complex transformations into simpler components.
  • Eigenvectors point in the "natural" directions of a transformation where only stretching or compression occurs
  • Any scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue
  • The set of all eigenvectors for one eigenvalue (plus the zero vector) forms the eigenspace
  • Eigenvectors for distinct eigenvalues are always linearly independent
  • In physics, eigenvectors represent principal axes of stress, inertia, or vibration modes

To find an eigenvector, you first need the eigenvalue. Then solve the homogeneous system (A - lambda I)v = 0. For 2x2 matrices this always reduces to a single equation relating the two components of v.

You can also calculate changes using our Eigenvalue Calculator, Eigenbasis Calculator, Diagonalization Calculator or Null Space Calculator.

Frequently Asked Questions

Why are there infinitely many eigenvectors for one eigenvalue?

Because any nonzero scalar multiple of an eigenvector is also an eigenvector. If [1, 1] is an eigenvector, so is [2, 2], [100, 100], or [-3, -3]. The calculator returns one representative direction.

What if I enter a value that is not an eigenvalue?

The system (A - lambda I)v = 0 will only have the trivial solution v = 0 if lambda is not actually an eigenvalue. Use the eigenvalue calculator first to find valid eigenvalues, then use this tool.

How do I find the eigenvector for a complex eigenvalue?

The same formula applies: solve (A - lambda I)v = 0 using complex arithmetic. The eigenvector components will be complex numbers. Complex eigenvectors always come in conjugate pairs.

What is the eigenspace?

The eigenspace for eigenvalue lambda is the set of all vectors v such that Av = lambda v, including the zero vector. It is a subspace of the vector space. For a 2x2 matrix, the eigenspace is either a line through the origin (1-dimensional) or the entire plane (2-dimensional, only when A = lambda I).

How are eigenvectors used in real applications?

In structural engineering, eigenvectors represent vibration modes. In computer vision, they define principal components of image data (eigenfaces). In quantum mechanics, they represent measurable states. In Google PageRank, the dominant eigenvector ranks web pages.

Can two different eigenvalues share the same eigenvector?

No. If Av = lambda1 v and Av = lambda2 v for the same nonzero v, then lambda1 must equal lambda2. Distinct eigenvalues always have linearly independent eigenvectors.

Related Calculators

Eigenvalue CalculatorEigenbasis CalculatorDiagonalization CalculatorNull Space Calculator

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