Eigenvalues and eigenvectors capture the intrinsic stretching/compressing directions of a linear transformation. For a square matrix $A$, a nonzero vector $v$ is an eigenvector with eigenvalue $\lambda$ if $Av = \lambda v$. Eigenvalues reveal fundamental properties: invertibility, stability, diagonalizability, and more.
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- Particular types of matricesParticular types of matrices are special matrices with specific structural properties or algebraic characteristics...
- Definition of eigenvalue of a matrixDefinition: A scalar $\lambda$ is an eigenvalue of an $n \times n$ matrix $A$ if there exists a nonzero vector $v$...
- Definition of eigenvector of a matrixDefinition: A nonzero vector $v$ is an eigenvector of a matrix $A$ corresponding to eigenvalue $\lambda$ if $Av =...
- EigenspacesDefinition: The eigenspace $E \lambda$ for eigenvalue $\lambda$ is the set of all eigenvectors corresponding to...
- Every matrix has an eigenvalue over the complex numbers.Theorem: Every square matrix over $\mathbb{C}$ has at least one eigenvalue. This follows from the Fundamental Theorem...
- Eigenvalues and operations on matricesProperties of eigenvalues under matrix operations: - If $\lambda$ is an eigenvalue of $A$, then $\lambda^k$ is an...
- Eigenvectors with distinct eigenvalues are linearly independent.Theorem: Eigenvectors corresponding to distinct eigenvalues are linearly independent. If $v 1, \ldots, v k$ are...
- MultiplicityDefinition: The algebraic multiplicity of an eigenvalue $\lambda$ is its multiplicity as a root of the characteristic...
- Characteristic and minimal polynomialsFor a square matrix use the monic convention $p A(t)=\det(tI-A)$. Its degree is the matrix size, and its roots are...
- The dimension of a eigenspace is less than or equal to the (algebraic) multiplicity of the eigenvalue.For an eigenvalue $\lambda$, $1\le\dim\ker(A-\lambda I)\le m \lambda$, where $m \lambda$ is its algebraic...
- Definition of eigenvalue/characteristic value of a linear transformationFor a linear operator $T:V\to V$ over $\mathbb F$, a scalar $\lambda\in\mathbb F$ is an eigenvalue if some nonzero...
- Definition of eigenvector/characteristic vector of a linear transformationA nonzero vector $v$ is an eigenvector of $T:V\to V$ for $\lambda$ if $T(v)=\lambda v$. Zero is excluded because it...
- Definition of characteristic polynomial of a linear transformationFor an operator $T$ on an $n$-dimensional space, define $p T(t)=\det(tI-[T] B)$ using any basis $B$. Similarity...
- Definition of minimal polynomial of a linear transformationThe minimal polynomial $m T$ of an operator on a finite-dimensional nonzero space is the monic polynomial of least...
- The Cayley-Hamilton theorem for a linear transformationCayley–Hamilton says $p T(T)=0$ for every finite-dimensional operator $T$, where $p T(t)=\det(tI-[T] B)$. Apply the...
- The minimal polynomial of a linear transformation exists and is unique.For finite-dimensional $V$, powers $I,T,T^2,\ldots$ are eventually dependent in the finite-dimensional space of...
- Definition of applying a polynomial to a linear transformationFor $p(t)=a 0+a 1t+\cdots+a kt^k$, define $p(T)=a 0I+a 1T+\cdots+a kT^k$ for an operator $T:V\to V$. The constant...
- Every nonzero finite-dimensional complex operator has an eigenvalueEvery operator on a nonzero finite-dimensional complex vector space has an eigenvalue, because its positive-degree...
- Definition of eigenspace of a linear transformationFor an eigenvalue $\lambda$, the eigenspace is $E \lambda(T)=\ker(T-\lambda I)$. It consists of zero and all...
- The eigenspace of a linear transformation is a nontrivial subspace.For eigenvalue $\lambda$, $E \lambda(T)$ is nonzero because an eigenvector exists. It is a subspace because it is the...
- Definition of invariant subspace of a linear transformation.A subspace $U\subseteq V$ is $T$-invariant if $T(U)\subseteq U$. Then the restriction $T U:U\to U$ is a well-defined...
- Invariant direct sums give block diagonal operator matricesIf $V=U 1\oplus\cdots\oplus U k$ and each $U i$ is invariant under $T$, concatenate bases of the components. The...
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