Eigenvalues and Eigenvectors

Linear algebra is the math of vectors and matrices. It lets you handle many equations, or many dimensions, all at the same time instead of one line at a time.

Eigenvalues and Eigenvectors: the central idea

An eigenvector is a nonzero direction that a linear transformation only scales. Its eigenvalue records the scaling factor, including reversal or collapse.

Words you need

Eigenvalue

An eigenvalue is the scaling factor lambda for which the equation Av = lambda v has a nonzero solution v, so it records how much the matrix stretches, shrinks, or flips one special direction.

Eigenvector

An eigenvector is a nonzero vector that a matrix maps onto a multiple of itself, meaning the transformation leaves its direction alone.

Characteristic polynomial

The characteristic polynomial is what you get by expanding the determinant of A minus lambda times the identity, and its roots are exactly the eigenvalues of A.

Eigenspace

The eigenspace of an eigenvalue is the whole collection of vectors the matrix scales by that eigenvalue, together with the zero vector, and it is the null space of A minus lambda times the identity.

Algebraic multiplicity

The algebraic multiplicity of an eigenvalue is how many times it appears as a root of the characteristic polynomial.

Geometric multiplicity

The geometric multiplicity of an eigenvalue is the number of independent eigenvectors it actually supplies, and it can be smaller than the algebraic multiplicity.

What to know before this lesson

Know matrix multiplication, determinants, homogeneous systems, null spaces, and polynomial roots.

If one of those prerequisites is uncertain, use the Linear Algebra subject guide to locate the earlier concept before memorizing a procedure.

Eigenvalues and eigenvectors: a worked example

Every step, with the arithmetic

1. Step 1 - Write A and subtract lambda down the diagonalA = [4 1 ; 2 3], so A - lambda I = [4-lambda 1 ; 2 3-lambda]
2. Step 2 - Take the determinant of that(4 - lambda)(3 - lambda) - (1)(2) = 12 - 7 lambda + lambda^2 - 2
3. Step 3 - Write the characteristic equationlambda^2 - 7 lambda + 10 = 0
4. Step 4 - Factor and read off the eigenvalues(lambda - 5)(lambda - 2) = 0, so lambda = 5 and lambda = 2
5. Step 5 - Find the eigenvector for lambda = 5A - 5I = [-1 1 ; 2 -2], and row one says -x + y = 0, so y = x and v1 = (1, 1)
6. Step 6 - Verify that eigenvectorA v1 = (4(1) + 1(1), 2(1) + 3(1)) = (5, 5) = 5(1, 1), correct
7. Step 7 - Find and verify the eigenvector for lambda = 2A - 2I = [2 1 ; 2 1], so 2x + y = 0 and v2 = (1, -2); check A v2 = (4 - 2, 2 - 6) = (2, -4) = 2(1, -2)
8. Step 8 - Use the 2 by 2 shortcut as a checkthe eigenvalues must add to the trace and multiply to the determinant: 5 + 2 = 7 = 4 + 3, and 5 x 2 = 10 = (4)(3) - (1)(2)

The characteristic equation identifies $\lambda$ values for which $A-\lambda I$ is singular; its nonzero null-space vectors are the corresponding eigenvectors.

Eigenvalues of common 2 by 2 matrices, with the trace and determinant you can check them against

The step-by-step method for eigenvalues and eigenvectors

1. Solve $\det(A-\lambda I)=0$ to find candidate eigenvalues.
2. For each $\lambda$, row-reduce $A-\lambda I$ and find a nonzero vector in its null space.
3. Verify $Av=\lambda v$ directly and distinguish algebraic from geometric multiplicity when needed.

How to check your answer

Multiply the original matrix by each proposed vector and compare every component with $\lambda v$.

Multiply back, inspect dimensions, and test the result on a simple vector. Solutions to a system must satisfy every original equation, not merely the reduced matrix.

A mistake that changes the mathematics

The zero vector satisfies $A0=\lambda0$ for every $\lambda$, so it is deliberately excluded from the definition.

Where you will actually use this

Search ranking

Google's original PageRank treats the whole web as one enormous matrix and takes the eigenvector belonging to eigenvalue 1 as the steady-state importance score of every page.

Bridges and buildings

Engineers compute the eigenvalues of a structure's stiffness matrix to find its natural vibration frequencies, then design so that wind or traffic never drives it at one of those resonant modes.

Principal component analysis

Compressing data means taking the eigenvectors of the covariance matrix with the largest eigenvalues, since those directions carry the most variation and the rest can be dropped.

How eigenvalues and eigenvectors connects to the rest of linear algebra

• Quadratic formula — The characteristic equation of a 2 by 2 matrix is a quadratic, so the quadratic formula is what produces the eigenvalues when the polynomial refuses to factor.
• Matrix operations — Checking an eigenpair is an ordinary matrix-times-vector multiplication, and diagonalizing means writing A as a product of three matrices.
• Solving linear systems — Every eigenvector comes from row-reducing the homogeneous system (A minus lambda I)v = 0, where the free variable is what lets you pick a convenient scale.

Try a transfer problem

Find the eigenpairs of $\begin{bmatrix}2&1\\0&2\end{bmatrix}$ and decide whether it has enough eigenvectors to diagonalize.

Show the worked answer

For A = [2 1 ; 0 2], A - lambda I = [2-lambda 1 ; 0 2-lambda], and because the matrix is triangular the determinant is just the diagonal product: (2 - lambda)(2 - lambda) - (1)(0) = (2 - lambda)^2. Setting that to zero gives lambda = 2 as a double root, so 2 has algebraic multiplicity 2. Now find the eigenvectors: A - 2I = [0 1 ; 0 0]. The one useful row says 0x + 1y = 0, so y = 0 while x is free. Every eigenvector has the form t(1, 0), and a check confirms it: A(1, 0) = (2(1) + 1(0), 0(1) + 2(0)) = (2, 0) = 2(1, 0). The eigenspace is only a single line, so the geometric multiplicity is 1 while the algebraic multiplicity is 2. A 2 by 2 matrix needs two independent eigenvectors to be diagonalized, and this one supplies only one. So A is not diagonalizable; it is what is called a defective matrix, which is typical of shear-style matrices.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to linear algebra practice questions.

Questions about eigenvalues and eigenvectors

Why is the zero vector not allowed to be an eigenvector?

Because A times the zero vector equals lambda times the zero vector for absolutely every number lambda, so if it counted, every number would be an eigenvalue of every matrix and the idea would carry no information. Eigenvalues are still allowed to be 0; it is the vector that must be nonzero.

Can an eigenvalue be zero, and what does that mean?

Yes. An eigenvalue of 0 means some nonzero direction gets crushed to the zero vector, so the matrix loses information and cannot be undone. It happens exactly when the determinant is 0, which is the same as saying the matrix has no inverse. The flattening matrix [1 0 ; 0 0] has eigenvalues 1 and 0 for this reason.

What does it mean when the eigenvalues come out complex?

It means no real direction is preserved, because the transformation includes a rotation. The quarter-turn matrix [0 -1 ; 1 0] gives lambda squared plus 1 equals 0, so the eigenvalues are the imaginary numbers i and -i. The size of a complex eigenvalue still tells you the scaling, and its angle tells you how far the rotation turns.

Is there a quick way to get the eigenvalues of a 2 by 2 matrix?

Yes. The two eigenvalues always add up to the trace, the sum of the diagonal entries, and multiply to the determinant. For [4 1 ; 2 3] the trace is 7 and the determinant is 10, so you need two numbers adding to 7 and multiplying to 10, which are 5 and 2. It is a fast check even when you solved the characteristic equation the long way.