Operators on Complex Vector Spaces

8A. Generalized Eigenvectors and Nilpotent Operators

1) Null Spaces of Powers

Increasing Null Spaces

For $T\in\mathcal{L}(V)$,

$$\{0\} = \operatorname{null}T^0 \subseteq \operatorname{null}T \subseteq \operatorname{null}T^2 \subseteq \cdots.$$

Meaning: Applying higher powers of $T$ can only increase the set of vectors that are sent to zero.

Stabilization of Null Spaces

If

$$\operatorname{null}T^m = \operatorname{null}T^{m+1},$$

then

$$\operatorname{null}T^m = \operatorname{null}T^{m+1} = \operatorname{null}T^{m+2} = \cdots.$$

Meaning: Once consecutive null spaces become equal, they never grow again.

Dimension Bound

If

$$n=\dim V,$$

then

$$\operatorname{null}T^n = \operatorname{null}T^{n+1} = \cdots.$$

Meaning: The null spaces must stop growing by the power $\dim V$.

2) Null-Range Decomposition

Power Decomposition

If

$$n=\dim V,$$

then

$$V = \operatorname{null}T^n \oplus \operatorname{range}T^n.$$

Meaning: A sufficiently large power separates $V$ into vectors eventually sent to zero and vectors remaining in the range.

3) Generalized Eigenvector

Generalized Eigenvector

Suppose $\lambda$ is an eigenvalue of $T$.

A nonzero vector $v\in V$ is a generalized eigenvector corresponding to $\lambda$ if

$$(T-\lambda I)^k v=0$$

for some positive integer $k$.

Meaning: A generalized eigenvector may not be an eigenvector immediately, but repeated application of $T-\lambda I$ eventually sends it to zero.

Dimension Criterion

A nonzero vector $v$ is a generalized eigenvector corresponding to $\lambda$ if and only if

$$(T-\lambda I)^{\dim V}v=0.$$

Meaning: It is enough to test the power $\dim V$.

4) Basis of Generalized Eigenvectors

Generalized Eigenvector Basis

If

$$\mathbb{F}=\mathbb{C},$$

then every operator $T\in\mathcal{L}(V)$ has a basis consisting of generalized eigenvectors.

Meaning: Even when ordinary eigenvectors do not form a basis, generalized eigenvectors always do over $\mathbb{C}$.

5) Independence of Generalized Eigenvectors

Distinct Eigenvalues

Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent.

Meaning: Generalized eigenvectors associated with different eigenvalues represent independent directions.

6) Nilpotent Operator

Nilpotent

An operator $T$ is nilpotent if

$$T^m=0$$

for some positive integer $m$.

Meaning: Repeated application of a nilpotent operator eventually sends every vector to zero.

Dimension Bound

If $T$ is nilpotent, then

$$T^{\dim V}=0.$$

Meaning: A nilpotent operator never needs a power larger than the dimension of the space to become zero.

7) Eigenvalues of a Nilpotent Operator

Nilpotent Eigenvalues

If $T$ is nilpotent, then its only eigenvalue is

$$0.$$

Meaning: A nilpotent operator cannot scale an eigenvector by a nonzero scalar.

Complex Converse

If $\mathbb{F}=\mathbb{C}$ and $0$ is the only eigenvalue of $T$, then $T$ is nilpotent.

Meaning: Over $\mathbb{C}$, having only eigenvalue zero completely characterizes nilpotent operators.

8) Minimal Polynomial of a Nilpotent Operator

Nilpotent Criterion

The following are equivalent:

$$T\text{ is nilpotent},$$ $$p_{\min}(z)=z^m$$

for some positive integer $m$, and $T$ has an upper-triangular matrix whose diagonal entries are all zero.

Meaning: Nilpotence can be recognized from powers, the minimal polynomial, or an upper-triangular representation.

8B. Generalized Eigenspace Decomposition

1) Generalized Eigenspace

Generalized Eigenspace

For $\lambda\in\mathbb{F}$,

$$G(\lambda,T) = \{v\in V: (T-\lambda I)^k v=0 \text{ for some positive integer }k\}.$$

Meaning: The generalized eigenspace contains the zero vector and all generalized eigenvectors associated with $\lambda$.

Eigenspace Inclusion

$$E(\lambda,T) \subseteq G(\lambda,T).$$

Meaning: Every ordinary eigenvector is automatically a generalized eigenvector.

2) Description of a Generalized Eigenspace

Null-Space Formula

$$G(\lambda,T) = \operatorname{null}(T-\lambda I)^{\dim V}.$$

Meaning: A generalized eigenspace is the stabilized null space of a power of $T-\lambda I$.

3) Generalized Eigenspace Decomposition

Generalized Eigenspace Decomposition

Suppose

$$\mathbb{F}=\mathbb{C}$$

and $\lambda_1,\ldots,\lambda_m$ are the distinct eigenvalues of $T$.

Then

$$V = G(\lambda_1,T) \oplus \cdots \oplus G(\lambda_m,T).$$

Meaning: Every complex finite-dimensional vector space splits into independent generalized eigenspaces of $T$.

4) Invariance of Generalized Eigenspaces

Invariant Subspace

Each

$$G(\lambda_k,T)$$

is invariant under $T$.

Meaning: Applying $T$ to a generalized eigenvector keeps it inside the same generalized eigenspace.

5) Nilpotent Part on a Generalized Eigenspace

Nilpotent Restriction

On $G(\lambda,T)$,

$$(T-\lambda I)|_{G(\lambda,T)}$$

is nilpotent.

Meaning: On each generalized eigenspace, $T$ behaves like a scalar multiple of the identity plus a nilpotent operator.

6) Multiplicity

Multiplicity

The multiplicity of an eigenvalue $\lambda$ is

$$\dim G(\lambda,T).$$

Equivalently,

$$\dim\operatorname{null}(T-\lambda I)^{\dim V}.$$

Meaning: Multiplicity measures the total dimension associated with an eigenvalue, including generalized eigenvectors.

7) Algebraic and Geometric Multiplicity

Algebraic Multiplicity

$$\dim G(\lambda,T).$$

Meaning: Algebraic multiplicity counts all generalized eigenvector directions associated with $\lambda$.

Geometric Multiplicity

$$\dim E(\lambda,T).$$

Meaning: Geometric multiplicity counts only ordinary eigenvector directions.

Multiplicity Relation

$$\dim E(\lambda,T) \le \dim G(\lambda,T).$$

Meaning: Geometric multiplicity cannot exceed algebraic multiplicity.

8) Sum of Multiplicities

Multiplicity Sum

If $\mathbb{F}=\mathbb{C}$, then

$$\sum_{\lambda} \dim G(\lambda,T) = \dim V.$$

Meaning: The multiplicities of all distinct eigenvalues add up to the dimension of the space.

9) Characteristic Polynomial

Characteristic Polynomial

Suppose the distinct eigenvalues of $T$ are

$$\lambda_1,\ldots,\lambda_m$$

with multiplicities

$$d_1,\ldots,d_m.$$

The characteristic polynomial is

$$q(z) = (z-\lambda_1)^{d_1} \cdots (z-\lambda_m)^{d_m}.$$

Meaning: The characteristic polynomial records every eigenvalue together with its multiplicity.

Degree

$$\deg q = \dim V.$$

Meaning: The degree of the characteristic polynomial equals the dimension of the vector space.

Zeros

The zeros of the characteristic polynomial are exactly the eigenvalues of $T$.

Meaning: The characteristic polynomial contains all eigenvalue information.

10) Cayley–Hamilton Theorem

Cayley–Hamilton Theorem

If $q$ is the characteristic polynomial of $T$, then

$$q(T)=0.$$

Meaning: Every operator satisfies its own characteristic polynomial.

11) Characteristic and Minimal Polynomials

Divisibility

The characteristic polynomial is a polynomial multiple of the minimal polynomial.

Thus,

$$p_{\min}(z)\mid q(z).$$

Meaning: The minimal polynomial contains only the factors and powers actually needed to annihilate $T$.

12) Multiplicity and Upper-Triangular Matrices

Diagonal Multiplicity

If $T$ has an upper-triangular matrix, the number of times an eigenvalue $\lambda$ occurs on the diagonal equals its multiplicity.

Meaning: The diagonal of an upper-triangular representation lists eigenvalues according to their algebraic multiplicities.

13) Block Diagonal Matrix

Block Diagonal Matrix

A block diagonal matrix has the form

$$\begin{pmatrix} A_1 & & 0\\ & \ddots &\\ 0 & & A_m \end{pmatrix},$$

where each $A_k$ is a square matrix.

Meaning: A block diagonal matrix separates a transformation into independent smaller transformations.

14) Block Form from Generalized Eigenspaces

Generalized Eigenspace Block Form

For a complex vector space, there exists a basis such that

$$\mathcal{M}(T) = \begin{pmatrix} A_1 & & 0\\ & \ddots &\\ 0 & & A_m \end{pmatrix},$$

where each $A_k$ is upper triangular with diagonal entries all equal to $\lambda_k$.

Meaning: Each block represents the action of $T$ on one generalized eigenspace.

8C. Consequences of Generalized Eigenspace Decomposition

1) Square Root of Identity Plus Nilpotent

Nilpotent Square Root

If $N$ is nilpotent, then

$$I+N$$

has a square root.

Meaning: Nilpotence makes the formal square-root expansion terminate after finitely many terms.

2) Square Roots of Invertible Complex Operators

Square Root Theorem

If $V$ is a complex vector space and $T$ is invertible, then $T$ has a square root.

Thus there exists $R\in\mathcal{L}(V)$ such that

$$R^2=T.$$

Meaning: Every invertible operator on a finite-dimensional complex vector space can be square-rooted.

3) Jordan Block

Jordan Block

A Jordan block corresponding to $\lambda$ has the form

$$\begin{pmatrix} \lambda & 1 & 0 & \cdots & 0\\ 0 & \lambda & 1 & \ddots & \vdots\\ \vdots & \ddots & \ddots & \ddots & 0\\ 0 & \cdots & 0 & \lambda & 1\\ 0 & \cdots & \cdots & 0 & \lambda \end{pmatrix}.$$

Meaning: A Jordan block consists of one eigenvalue on the diagonal and possibly a chain of generalized eigenvectors represented by the ones above the diagonal.

4) Jordan Basis

Jordan Basis

A basis of $V$ is a Jordan basis for $T$ if the matrix of $T$ is block diagonal and every block is a Jordan block.

Meaning: A Jordan basis arranges generalized eigenvectors into chains that reveal the structure of the operator.

5) Jordan Basis for Nilpotent Operators

Nilpotent Jordan Form

Every nilpotent operator has a Jordan basis.

For a nilpotent operator, each Jordan block has the form

$$\begin{pmatrix} 0 & 1 & & 0\\ & 0 & \ddots &\\ & & \ddots & 1\\ 0 & & & 0 \end{pmatrix}.$$

Meaning: A nilpotent operator consists entirely of generalized eigenvector chains associated with eigenvalue zero.

6) Jordan Form

Jordan Form

If

$$\mathbb{F}=\mathbb{C},$$

then every operator $T\in\mathcal{L}(V)$ has a Jordan basis.

Meaning: Every finite-dimensional complex operator can be represented by a block diagonal matrix made of Jordan blocks.

7) Structure of a Jordan Block

Scalar Plus Nilpotent

A Jordan block can be written as

$$J_\lambda = \lambda I+N,$$

where $N$ is nilpotent.

Meaning: Every Jordan block separates into ordinary eigenvalue scaling plus a nilpotent part.

8) Diagonalizable Operators and Jordan Form

Diagonalizable Case

$T$ is diagonalizable exactly when every Jordan block has size $1$.

Meaning: Larger Jordan blocks measure the failure of an operator to have enough ordinary eigenvectors.

8D. Trace: A Connection Between Matrices and Operators

1) Trace of a Matrix

Trace

For a square matrix $A$,

$$\operatorname{tr}A$$

is the sum of its diagonal entries.

If

$$A=(A_{j,k}),$$

then

$$\operatorname{tr}A = \sum_{k=1}^{n}A_{k,k}.$$

Meaning: The trace extracts one scalar by adding all diagonal entries.

2) Trace of a Product

Cyclic Property

If the products are defined, then

$$\operatorname{tr}(AB) = \operatorname{tr}(BA).$$

Meaning: Although matrix multiplication is not commutative, swapping two factors does not change the trace.

3) Basis Independence

Trace Does Not Depend on Basis

If two matrices represent the same operator using different bases, they have the same trace.

Meaning: Trace belongs to the operator itself, not to a particular matrix representation.

4) Trace of an Operator

Operator Trace

For $T\in\mathcal{L}(V)$,

$$\operatorname{tr}T = \operatorname{tr}\mathcal{M}(T),$$

where any basis of $V$ may be used.

Meaning: The trace of an operator is the trace of any matrix representing it.

5) Trace and Eigenvalues

Sum of Eigenvalues

If

$$\mathbb{F}=\mathbb{C},$$

then

$$\operatorname{tr}T$$

equals the sum of the eigenvalues of $T$, counted according to their multiplicities.

If the eigenvalues are

$$\lambda_1,\ldots,\lambda_n,$$

then

$$\operatorname{tr}T = \lambda_1+\cdots+\lambda_n.$$

Meaning: Trace measures the total eigenvalue contribution of an operator.

6) Trace and Characteristic Polynomial

Characteristic Polynomial Coefficient

If

$$q(z) = z^n+c_{n-1}z^{n-1}+\cdots,$$

is the characteristic polynomial of $T$, then

$$\operatorname{tr}T = -c_{n-1}.$$

Meaning: The trace appears directly as the negative of the second-highest coefficient of the characteristic polynomial.

7) Trace on an Inner Product Space

Orthonormal Basis Formula

If

$$e_1,\ldots,e_n$$

is an orthonormal basis, then

$$\operatorname{tr}T = \langle Te_1,e_1\rangle +\cdots+ \langle Te_n,e_n\rangle.$$

Meaning: In an orthonormal basis, trace is the sum of the diagonal inner-product components.

8) Linearity of Trace

Addition

$$\operatorname{tr}(S+T) = \operatorname{tr}S + \operatorname{tr}T.$$

Scalar Multiplication

$$\operatorname{tr}(\lambda T) = \lambda\operatorname{tr}T.$$

Meaning: Trace is a linear functional on $\mathcal{L}(V)$.

9) Cyclic Property for Operators

Trace of Products

$$\operatorname{tr}(ST) = \operatorname{tr}(TS).$$

Meaning: The trace is unchanged when two operator factors are cyclically exchanged.

10) Trace of a Commutator

Commutator Trace

$$\operatorname{tr}(ST-TS)=0.$$

Meaning: Every commutator of finite-dimensional operators has trace zero.

Identity Is Not a Commutator

There do not exist $S,T\in\mathcal{L}(V)$ such that

$$ST-TS=I.$$

Meaning: The identity has nonzero trace, while every commutator has trace zero.