Eigenvalues & eigenvectors
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Flip to reveal answersWhat is an eigenvector of a matrix A?
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All 8 Flashcards — Eigenvalues & eigenvectors
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Question
What is an eigenvector of a matrix A?
Answer
A non-zero direction v that A only stretches: Av = λv (same direction, scaled by the eigenvalue λ).
Question
How do you find the eigenvalues of A?
Answer
Solve the characteristic equation det(A − λI) = 0 for λ (subtract λ down the diagonal, set the determinant to zero).
Question
How do you find an eigenvector for a given eigenvalue λ?
Answer
Solve (A − λI)v = 0; the rows give one line, so pick the simplest non-zero whole-number vector on it.
Question
In A = PDP⁻¹, what are P and D?
Answer
P has the eigenvectors as its columns; D has the matching eigenvalues on its diagonal (same order).
Question
How do you compute a power Aⁿ once A is diagonalised?
Answer
Aⁿ = PDⁿP⁻¹, and Dⁿ just raises each diagonal eigenvalue to the power n.
Question
For a transition matrix, what does the eigenvalue λ = 1 give you?
Answer
Its eigenvector is the steady state — the long-run mix the matrix leaves unchanged (rescale so the entries sum to the total).
Question
What happens to the part of a state along an eigenvector with |λ| < 1 as time goes on?
Answer
It is multiplied by λⁿ → 0, so that part fades away, leaving only the λ = 1 piece.
Question
Eigenvalues of a triangular (or diagonal) matrix?
Answer
They are exactly the entries on its main diagonal.
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Eigenvalues & eigenvectors (HL only)
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