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Eigenvalues and eigenvectors

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Particular types of matrices
Definition of eigenvector of a matrix
Eigenspaces
Every matrix has an eigenvalue over the complex numbers.
Multiplicity
Characteristic and minimal polynomials
Definition of eigenvalue/characteristic value of a linear transformation
Definition of eigenspace of a linear transformation
Definition of eigenvalue of a matrix
Eigenvalues and operations on matrices
Eigenvectors with distinct eigenvalues are linearly independent.
The dimension of a eigenspace is less than or equal to the (algebraic) multiplicity of the eigenvalue.
Definition of eigenvector/characteristic vector of a linear transformation
Definition of characteristic polynomial of a linear transformation
Definition of minimal polynomial of a linear transformation
The Cayley-Hamilton theorem for a linear transformation
The minimal polynomial of a linear transformation exists and is unique.
Definition of applying a polynomial to a linear transformation
Every nonzero finite-dimensional complex operator has an eigenvalue
The eigenspace of a linear transformation is a nontrivial subspace.
Definition of invariant subspace of a linear transformation.
Invariant direct sums give block diagonal operator matrices

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The dimension of a eigenspace is less than or equal to the (algebraic) multiplicity of the eigenvalue.

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Diagonalizability requires splitting and equality of geometric and algebraic multiplicitiesClose match
Definition of geometric multiplicity of an eigenvalueClose match
An eigenspace of a matrix is a nontrivial subspace.Close match
The eigenvalues of a matrix are the roots/solutions of its characteristic polynomial/equation.Close match
MultiplicityClose match
Definition of (algebraic) multiplicity of an eigenvalueClose match
An n-by-n matrix with n distinct eigenvalues is diagonalizable.Close match
An n-by-n matrix is diagonalizable if and only if the sum of the dimensions of the eigenspaces equals n.Close match
Definition of eigenspace of a linear transformationClose match
Every matrix has an eigenvalue over the complex numbers.Close match
Every nonzero finite-dimensional complex operator has an eigenvalueClose match
An n-by-n complex matrix has n eigenvalues counting multiplicityClose match
Definition of eigenvalue of a matrixClose match
EigenspacesClose match
The eigenvalues of a scalar multiple of a matrix are the scalar multiples of the eigenvalues.Close match
The eigenspace of a linear transformation is a nontrivial subspace.Close match
Eigenvalues and operations on matricesClose match
A matrix with real entries has eigenvalues occurring in conjugate pairs.Close match
Eigenvectors with distinct eigenvalues are linearly independent.Related
Definition of eigenvalue/characteristic value of a linear transformationRelated

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