Master Eigenvalues and Eigenvectors: 29 Flashcards from the Characteristic Polynomial to PageRank
Master eigenvalues, eigenvectors, diagonalization and the spectral theorem with 29 real flashcards, plus the trace trick for checking 2x2 answers fast.
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Deck ZX5976 on Flip It packs eigenvalues and eigenvectors into 29 cards — short, well written, and entirely conceptual. There is not a single matrix to invert or determinant to expand in the whole deck. That makes it excellent for the two days before an exam and useless on its own for building computational speed, which is worth knowing before you start.
Study this deck: Eigenvalues & Eigenvectors
1 / 10Eigenvector of a matrix A
Tap the card for the answer · swipe right if you knew it, left if not
Nine cards that carry the deck
| Front | Back |
|---|---|
| Find eigenvalues of a matrix A | Solve the characteristic equation det(A − λI) = 0, where I is the identity matrix |
| Characteristic polynomial | The polynomial det(A − λI); its roots are the eigenvalues of matrix A |
| Geometric multiplicity of an eigenvalue | The dimension of the eigenspace (nullspace of A − λI); number of linearly independent eigenvectors for that eigenvalue |
| Eigenvalues of a triangular matrix | The diagonal entries of the matrix |
| If A has eigenvalue λ, what are eigenvalues of A⁻¹ (if invertible)? | 1/λ (with same eigenvectors); A is invertible iff all eigenvalues are non-zero |
| Spectral Theorem for symmetric matrices | Every real symmetric matrix is orthogonally diagonalizable: A = QΛQ^T, Q orthogonal, Λ diagonal with eigenvalues |
| Defective matrix | A matrix that is not diagonalizable; fewer than n linearly independent eigenvectors |
| Gershgorin's Circle Theorem | Every eigenvalue lies within at least one disc centred at a diagonal entry, with radius equal to the sum of the absolute off-diagonal entries in that row |
| How are eigenvalues used in Google's PageRank? | Web pages are ranked by the dominant eigenvector of the link matrix |
The full map of what you will know
Eigenvector and eigenvalue from the defining equation; the characteristic polynomial; algebraic versus geometric multiplicity and the inequality between them; eigenspaces; diagonalization as A = PDP⁻¹; trace as the sum of eigenvalues and determinant as their product; eigenvalues of triangular, identity, zero, symmetric and orthogonal matrices; what happens to eigenvalues under powers and inversion; the spectral theorem; defective matrices; similar matrices sharing a spectrum; the power method; Gershgorin discs; and the applications — PCA and covariance matrices, PageRank, natural frequencies in vibration analysis, quantum energy levels, and the stability of dy/dt = Ay through solutions in e^(λt).
Turn the trace and determinant cards into a checking tool
The two cards giving tr(A) = Σλᵢ and det(A) = Πλᵢ look like trivia. They are your fastest error check, and the deck never says so. For any 2×2 matrix, the eigenvalues are the two numbers whose sum is the trace and whose product is the determinant — so the characteristic polynomial is λ² − tr(A)λ + det(A), and you can read the eigenvalues of [[3,1],[0,2]] or [[2,1],[1,2]] straight off. Since the deck gives you no matrices at all, write three of your own on paper — one triangular, one symmetric, one with a repeated eigenvalue — and solve them between passes. The repeated case is what exposes whether you really understand the defective-matrix card.
Read the truncated fronts as prompts
Several fronts lost their question word when the deck was built: Eigenvector of a matrix A, A matrix to be diagonalizable, Similar matrices and how do their eigenvalues relate. Mentally prefix them with "Define" or "When is" and they work fine. One card needs more care: it states that geometric multiplicity ≤ algebraic multiplicity with equality iff the matrix is diagonalizable "for that eigenvalue". True per eigenvalue, but the theorem you will be graded on is global — A is diagonalizable only when equality holds for every eigenvalue at once. Add that sentence to the back before your first pass.
Deck ZX5976 — Eigenvalues & Eigenvectors is in your library on Flip It.
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