Eigenvalues, eigenvectors, and diagonalization from a college linear algebra course.
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- What is an eigenvalue of a square matrix A?
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A scalar λ such that Av = λv for some non-zero vector v.- What is an eigenvector of a square matrix A?
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A non-zero vector v such that Av = λv for some scalar λ.- What is the characteristic polynomial of a matrix A?
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The polynomial det(A - λI), where λ is a variable and I is the identity matrix.- What is the characteristic equation of a matrix A?
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The equation det(A - λI) = 0, whose solutions are the eigenvalues of A.- How do you find the eigenvalues of a matrix A?
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Solve the characteristic equation det(A - λI) = 0.- What is an eigenspace of A corresponding to eigenvalue λ?
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The set of all vectors v such that (A - λI)v = 0, which is the null space of A - λI.- How do you find eigenvectors for a given eigenvalue λ?
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Solve the homogeneous system (A - λI)x = 0.- Can the zero vector be an eigenvector?
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No, eigenvectors are defined to be non-zero vectors.- What is the algebraic multiplicity of an eigenvalue?
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Its multiplicity as a root of the characteristic polynomial.- What is the geometric multiplicity of an eigenvalue?
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The dimension of the eigenspace, or the number of linearly independent eigenvectors for that eigenvalue.- What is the relationship between algebraic and geometric multiplicity?
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Geometric multiplicity is always less than or equal to algebraic multiplicity.- What is a diagonalizable matrix?
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A matrix that is similar to a diagonal matrix, or can be written as A = PDP^(-1) where D is diagonal and P is invertible.- When is an n by n matrix diagonalizable?
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When it has n linearly independent eigenvectors.- If a matrix has n distinct eigenvalues, what can you conclude?
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The matrix is diagonalizable, because eigenvectors for distinct eigenvalues are linearly independent.- What are the columns of the matrix P in A = PDP^(-1)?
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Linearly independent eigenvectors of A, with the order matching the eigenvalues on the diagonal of D.
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