What does the trace tell you?

The sum of eigenvalues equals the trace (a+d). The product of eigenvalues equals the determinant. These are quick sanity checks.

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Eigenvalue Calculator

Q: What is an eigenvalue?

A scalar lambda where Av = lambda*v for a non-zero vector v. It describes how much the matrix stretches its eigenvector.

Q: How do you find eigenvalues of a 2x2 matrix?

Solve det(A - lambda*I) = 0, which gives lambda^2 - trace*lambda + det = 0. Use the quadratic formula with discriminant = trace^2 - 4*det.

Q: What is the characteristic polynomial?

det(A - lambda*I) = 0. For 2x2: lambda^2 - (a+d)*lambda + (ad-bc) = 0. Eigenvalues are its roots.

Q: Can eigenvalues be complex?

Yes. When discriminant (trace^2 - 4*det) < 0, eigenvalues are complex conjugate pairs. This indicates rotational behavior.

Q: What is an eigenvector?

A non-zero vector that only gets scaled (not rotated) by matrix A. Each eigenvalue has a corresponding eigenvector direction.

Q: What are eigenvalues used for?

Google PageRank, PCA in data science, quantum mechanics, structural vibration analysis, image compression, and stability analysis of dynamic systems.

Q: What is a repeated eigenvalue?

When discriminant = 0, both eigenvalues are equal. The matrix may be defective (fewer than 2 independent eigenvectors).

Find eigenvalues and eigenvectors of a 2x2 matrix with step-by-step characteristic polynomial derivation and full solution.

2x2 Matrix-Characteristic Polynomial-Eigenvectors-Step-by-Step

2x2 Matrix

Presets

Matrix A

[2 1]
[1 2]

Trace (a+d):4

Determinant:3

Discriminant:4

Results

Eigenvalue 1

λ1 = 3

Eigenvector (normalized):

v1 = [0.7071, 0.7071]

Eigenvalue 2

λ2 = 1

Eigenvector (normalized):

v2 = [0.7071, -0.7071]

Characteristic Polynomial

λ² - (4)λ + (3) = 0

Key Properties

Sum of eigenvalues = Trace = 4

Product of eigenvalues = Determinant = 3

Eigenvalue Examples for Common Matrix Types

Matrix Type	Matrix	Eigenvalues	Interpretation
Identity	[1 0; 0 1]	lambda = 1, 1	All vectors are eigenvectors
Diagonal	[3 0; 0 -2]	lambda = 3, -2	Read from diagonal directly
Rotation 90°	[0 -1; 1 0]	lambda = i, -i	Complex - no real eigenvectors
Shear	[1 1; 0 1]	lambda = 1, 1	Repeated eigenvalue
Symmetric	[4 2; 2 1]	lambda = 5, 0	Real eigenvalues guaranteed

Understanding Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are among the most powerful concepts in linear algebra. The word "eigen" comes from German and means "own" or "characteristic" - eigenvalues are the characteristic values that describe a matrix's fundamental behavior.

Think of a matrix as a transformation of space - it stretches, rotates, and shears vectors. Most vectors change both direction and magnitude when transformed. Eigenvectors are special: they only change magnitude (get scaled by the eigenvalue) but keep their direction. The eigenvalue tells you exactly how much scaling occurs - a positive eigenvalue greater than 1 stretches, between 0 and 1 compresses, negative flips direction.

Real-World Applications

Google's original PageRank algorithm finds the eigenvector of the web's link matrix - the eigenvector corresponding to the largest eigenvalue gives each page's rank. Principal Component Analysis (PCA) in data science finds the eigenvectors of a covariance matrix to identify the directions of greatest variance. Quantum mechanics uses eigenvalues to describe the measurable states of physical observables.

Frequently Asked Questions

What is an eigenvalue?⌄

An eigenvalue is a scalar that describes how much a matrix stretches or compresses its eigenvector. For a matrix A and vector v, if Av = lambda*v (where lambda is a scalar), then lambda is an eigenvalue and v is the corresponding eigenvector.

How do you find eigenvalues of a 2x2 matrix?⌄

Set up the characteristic equation: det(A - lambda*I) = 0. For a 2x2 matrix [a b; c d], this gives lambda^2 - (a+d)*lambda + (ad-bc) = 0. Solve this quadratic for lambda using the quadratic formula. The trace (a+d) and determinant (ad-bc) are key.

What is the characteristic polynomial?⌄

The characteristic polynomial is det(A - lambda*I) = 0. For a 2x2 matrix, it's the quadratic lambda^2 - trace*lambda + det = 0. The eigenvalues are the roots of this polynomial.

What do eigenvalues tell you about a matrix?⌄

Eigenvalues reveal fundamental properties: if any eigenvalue is 0, the matrix is singular (non-invertible). The determinant equals the product of eigenvalues. The trace equals their sum. Eigenvalues determine whether a system is stable, oscillating, or diverging.

Can eigenvalues be complex numbers?⌄

Yes. When the discriminant (trace^2 - 4*det) is negative, eigenvalues are complex conjugate pairs. This occurs with rotation matrices and indicates oscillatory behavior rather than pure stretching.

What is an eigenvector?⌄

An eigenvector is a non-zero vector that only gets scaled (not rotated) when multiplied by matrix A. If Av = lambda*v, then v is an eigenvector. Each eigenvalue has a corresponding eigenvector. Eigenvectors pointing in the same direction are considered equivalent.

What are eigenvalues used for in real life?⌄

Eigenvalues appear in PCA (data science), Google's PageRank algorithm, quantum mechanics, structural engineering (vibration modes), population modeling, image compression, and face recognition systems. They reveal the natural modes of a system.

What is a repeated eigenvalue?⌄

A repeated eigenvalue occurs when the discriminant equals zero, meaning the characteristic polynomial has a double root. The matrix may not have two independent eigenvectors in this case (defective matrix).

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