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