21. Eigenvalues and Eigenvectors

Overview

Eigenvalues describe how a square matrix scales vectors that remain on the same line: Ax = λx. The lecture develops the characteristic equation det(A − λI) = 0, then uses projections, a component-swap matrix, a symmetric 2×2 matrix, a 90-degree rotation, and a triangular matrix to show eigenvalues and eigenvectors can be real, complex, shifted by adding a scalar identity matrix, or insufficient to form a full independent basis.

Key takeaways

A nonzero vector x is an eigenvector of A exactly when Ax = λx; for λ = 0, x must be a nonzero member of the null space.
The characteristic equation det(A − λI) = 0 converts the eigenvalue problem into finding roots of a polynomial, after which eigenvectors are found by computing null spaces.
For a projection, vectors in the projected plane have eigenvalue 1 and vectors perpendicular to the plane have eigenvalue 0.
Adding cI to A leaves its eigenvectors unchanged and shifts each eigenvalue by c, but adding a general matrix B does not generally add the eigenvalues of A and B.
A real matrix can have complex eigenvalues: a 90-degree rotation has eigenvalues i and −i, while symmetric matrices have real eigenvalues.
Repeated eigenvalues do not guarantee enough independent eigenvectors: [[3, 1], [0, 3]] has eigenvalue 3 twice but only one independent eigenvector.

Chapters

0:00 Eigenvectors as Directions Preserved by a Matrix

An eigenvector x satisfies Ax = λx, so matrix multiplication changes its scale without changing its line.
The eigenvalue λ can be positive, negative, zero, or complex; most vectors are not eigenvectors.
If λ = 0, then Ax = 0, so nonzero eigenvectors with eigenvalue zero belong to the null space and exist when A is singular.

5:18 Projection Matrices Have Eigenvalues 1 and 0

A projection leaves every vector already in the target plane unchanged, making those vectors eigenvectors with eigenvalue 1.
Vectors perpendicular to the plane project to zero, so they are eigenvectors with eigenvalue 0 and lie in the null space.
A vector with both in-plane and perpendicular components generally changes direction when projected, so it is not an eigenvector.

10:32 A Component-Swap Matrix Reveals Eigenvalues ±1

For A = [[0, 1], [1, 0]], the vector (1, 1) is unchanged by swapping its components and has eigenvalue 1.
The vector (−1, 1) becomes its negative after the swap, so its eigenvalue is −1.
The two eigenvalues sum to the trace, which is 0; more generally, the sum of eigenvalues equals the sum of diagonal entries.

16:08 Deriving the Characteristic Equation

Rearranging Ax = λx gives (A − λI)x = 0, where I is the identity matrix.
A nonzero eigenvector exists only if A − λI is singular, which requires det(A − λI) = 0.
Solving this characteristic equation finds eigenvalues first; each eigenvector then comes from the null space of A − λI.

21:00 Finding Eigenvalues of a Symmetric 2×2 Matrix

For A = [[3, 1], [1, 3]], det(A − λI) = (3 − λ)² − 1 = λ² − 6λ + 8.
The quadratic factors as (λ − 4)(λ − 2), giving eigenvalues 4 and 2.
In this 2×2 case, the characteristic polynomial has the form λ² − (trace A)λ + det(A), with trace 6 and determinant 8.

26:00 Computing Eigenvectors and Shifting the Matrix by 3I

For eigenvalue 4, A − 4I has null space spanned by (1, 1); for eigenvalue 2, A − 2I has null space spanned by (−1, 1).
The eigenvectors (1, 1) and (−1, 1) are perpendicular, illustrating a characteristic property of symmetric matrices.
Adding 3I to a matrix preserves its eigenvectors and increases every eigenvalue by 3, as seen when eigenvalues 1 and −1 become 4 and 2.

33:17 Why Eigenvalues of A + B Usually Cannot Be Added

Knowing Ax = λx and that B has eigenvalue α does not imply Bx = αx, because B generally has different eigenvectors.
Therefore, the eigenvalues of A + B are not generally sums of eigenvalues of A and B; eigenvalues also do not generally multiply for AB.
The special case B = cI works because every vector is an eigenvector of cI, shifting eigenvalues by c without changing eigenvectors.

37:10 A 90-Degree Rotation Produces Complex Eigenvalues

The 90-degree rotation matrix Q = [[0, −1], [1, 0]] has trace 0 and determinant 1.
Its characteristic equation is λ² + 1 = 0, so its eigenvalues are i and −i rather than real numbers.
No nonzero real vector remains parallel to itself after a 90-degree rotation, but complex eigenvalues provide solutions to the eigenvalue equation.

41:17 Complex Conjugates and the Role of Symmetry

The rotation matrix's eigenvalues i and −i form a complex-conjugate pair, showing that real matrix entries do not guarantee real eigenvalues.
Symmetric matrices have real eigenvalues, while the rotation matrix is skew-symmetric because Qᵀ = −Q.
The contrast between symmetric and skew-symmetric matrices illustrates how a matrix's structure constrains its eigenvalues.

45:57 A Repeated Eigenvalue Can Have Too Few Eigenvectors

For the triangular matrix [[3, 1], [0, 3]], both eigenvalues are 3, readable from its diagonal and confirmed by det(A − λI) = (3 − λ)².
The null space of A − 3I is only the line spanned by (1, 0), so the repeated eigenvalue has just one independent eigenvector.
A 2×2 matrix can therefore fail to provide two independent eigenvectors, a defective case that prevents eigenvectors from forming a full basis.

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