Lecture 4: Eigenvalues and Eigenvectors

What this lecture covers

This lecture builds up eigenvalues and eigenvectors from scratch: what makes a vector an eigenvector, why they solve powers of a matrix and exponentials so easily, and how they let you write down the general solution of a difference equation or a linear differential equation. Strang also introduces similar matrices, M^-1 A M, and shows why matrices related this way share the same eigenvalues.

The lecture then narrows to symmetric matrices, the S in the course's running letter scheme. Strang works a small example to compute eigenvalues by hand, checks them with the trace and determinant, and derives the diagonalization A = X Λ X^-1, ending with the spectral theorem: a symmetric matrix's eigenvectors are orthogonal and its eigenvalues are real. After watching you should be able to compute eigenvalues and eigenvectors of a small matrix, explain why similar matrices share eigenvalues, and state the spectral theorem.

Key ideas

• Eigenvector definition: Ax = λx means multiplying by A doesn't change the direction of x, only scales it by λ.
• Powers and functions of A: if x is an eigenvector of A with eigenvalue λ, then x is also an eigenvector of A^k (eigenvalue λ^k), of A^-1 (eigenvalue 1/λ, when λ ≠ 0), and of the matrix exponential e^(At) (eigenvalue e^(λt)).
• Solving difference and differential equations: writing any starting vector as a combination of eigenvectors lets you solve v_(k+1) = A v_k or dv/dt = Av immediately, since each eigenvector component evolves independently.
• Similar matrices: B = M^-1 A M for any invertible M means A and B have the same eigenvalues; this is the basis for how eigenvalues are computed numerically, by repeatedly choosing good M's.
• AB and BA share eigenvalues: even though A and B are not similar to each other in general, AB and BA are similar (via M = B), so they share nonzero eigenvalues.
• Trace and determinant checks: for a small matrix, the sum of the eigenvalues equals the trace (sum of diagonal entries), and the product of the eigenvalues equals the determinant.
• Diagonalization: A = X Λ X^-1, where X has the eigenvectors as columns and Λ is diagonal with the eigenvalues; for a symmetric matrix this becomes S = Q Λ Q^T with Q orthogonal, the spectral theorem.

Walkthrough

What eigenvectors are good for (1:03)

Strang defines eigenvectors and eigenvalues for a general square matrix A, then shows that an eigenvector of A is automatically an eigenvector of A^2, A^k, A^-1 (when λ ≠ 0), and the matrix exponential, with the eigenvalue transformed accordingly (λ^2, λ^k, 1/λ, e^(λt)). Writing a starting vector v as a combination of eigenvectors then gives an immediate solution to a one-step difference equation v_(k+1) = A v_k or a differential equation dv/dt = Av.

Similar matrices and eigenvalue computation (12:21)

Strang defines similar matrices, B = M^-1 A M, and proves they share the same eigenvalues, with eigenvectors related by the same M. He notes this is the mechanism behind how software actually computes eigenvalues: applying a sequence of similarity transformations that gradually push a matrix toward triangular (or, for symmetric matrices, diagonal) form while the eigenvalues stay fixed. He also shows AB and BA share nonzero eigenvalues, while eigenvalues of A + B generally have no simple relation to the eigenvalues of A and B separately.

Computing eigenvalues of an antisymmetric matrix (25:40)

Before turning to symmetric matrices, Strang contrasts them with an antisymmetric 90-degree rotation matrix, showing it has no real eigenvector since rotation can never send a vector to a multiple of itself. Solving det(A - λI) = 0 gives complex eigenvalues i and -i, and Strang checks the result using the trace (sum of eigenvalues equals sum of diagonal entries) and the determinant (product of eigenvalues equals the determinant).

Symmetric matrices, diagonalization, and the spectral theorem (34:55)

Working a 2x2 symmetric example, Strang finds real eigenvalues 1 and -1 with orthogonal eigenvectors, checked again via trace and determinant. He builds the diagonalization A = X Λ X^-1, where X collects the eigenvectors as columns, and shows the same relation squares cleanly: A^2 = X Λ^2 X^-1. For a symmetric matrix, the eigenvector matrix becomes an orthogonal matrix Q, giving S = Q Λ Q^T, the spectral theorem, with real eigenvalues and orthogonal eigenvectors.

Before you watch

• Watch Lecture 3 on orthogonal matrices first, since the spectral theorem at the end of this lecture builds directly on Q^T Q = I.
• Comfort with determinants and solving det(A - λI) = 0 for a 2x2 matrix is assumed for the worked examples.

Check your understanding

1. Why is x also an eigenvector of A^2 and A^-1 whenever it is an eigenvector of A?
2. How does writing a vector as a combination of eigenvectors solve v_(k+1) = A v_k?
3. Why do A and B = M^-1 A M always have the same eigenvalues?
4. Why does an antisymmetric matrix like a 90-degree rotation have no real eigenvectors?
5. What does the spectral theorem S = Q Λ Q^T say about the eigenvalues and eigenvectors of a symmetric matrix?