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What is an eigenvector of a matrix A?
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All Flashcards in Topic 1.15
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1.15.18 cards
What is an eigenvector of a matrix A?
A non-zero direction v that A only stretches: Av = λv (same direction, scaled by the eigenvalue λ).
How do you find the eigenvalues of A?
Solve the characteristic equation det(A − λI) = 0 for λ (subtract λ down the diagonal, set the determinant to zero).
How do you find an eigenvector for a given eigenvalue λ?
Solve (A − λI)v = 0; the rows give one line, so pick the simplest non-zero whole-number vector on it.
In A = PDP⁻¹, what are P and D?
P has the eigenvectors as its columns; D has the matching eigenvalues on its diagonal (same order).
How do you compute a power Aⁿ once A is diagonalised?
Aⁿ = PDⁿP⁻¹, and Dⁿ just raises each diagonal eigenvalue to the power n.
For a transition matrix, what does the eigenvalue λ = 1 give you?
Its eigenvector is the steady state — the long-run mix the matrix leaves unchanged (rescale so the entries sum to the total).
What happens to the part of a state along an eigenvector with |λ| < 1 as time goes on?
It is multiplied by λⁿ → 0, so that part fades away, leaving only the λ = 1 piece.
Eigenvalues of a triangular (or diagonal) matrix?
They are exactly the entries on its main diagonal.
Topic 1.15 study notes
Full notes & explanations for Eigenvalues & eigenvectors (HL only)
Math AI exam skills
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