Eigenvalues and Eigenvectors

5A. Invariant Subspaces

1) Operator

Operator

A linear map from a vector space to itself is called an operator.

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

Meaning: An operator is a linear map whose domain and codomain are the same vector space.

2) Invariant Subspace

Invariant Subspace

A subspace $U$ of $V$ is invariant under $T\in\mathcal{L}(V)$ if

$$Tu\in U$$

for every $u\in U$.

Meaning: Applying $T$ to a vector in $U$ never moves the vector outside $U$.

Restriction to an Invariant Subspace

If $U$ is invariant under $T$, then

$$T|_U$$

is an operator on $U$.

Meaning: An invariant subspace can be studied independently using the restriction of $T$.

3) Eigenvalue

Eigenvalue

A scalar $\lambda\in\mathbb{F}$ is an eigenvalue of $T$ if there exists a nonzero vector $v\in V$ such that

$$Tv=\lambda v.$$

Meaning: An eigenvalue tells how $T$ scales some nonzero vector without changing its direction.

One-Dimensional Invariant Subspace

$T$ has an eigenvalue $\lambda$ with eigenvector $v$ exactly when

$$\operatorname{span}(v)$$

is invariant under $T$.

Meaning: Eigenvectors correspond to one-dimensional invariant subspaces.

4) Eigenvector

Eigenvector

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

$$Tv=\lambda v.$$

Equivalently,

$$v\in\operatorname{null}(T-\lambda I), \qquad v\neq0.$$

Meaning: An eigenvector is changed only by multiplication by a scalar.

5) Eigenvalue Criterion

Equivalent Conditions for an Eigenvalue

If $V$ is finite-dimensional, then

$$\lambda\text{ is an eigenvalue of }T$$

if and only if

$$T-\lambda I$$

is not injective.

Equivalently,

$$T-\lambda I$$

is not surjective or not invertible.

Meaning: Finding an eigenvalue is equivalent to finding a scalar that makes $T-\lambda I$ noninvertible.

6) Independent Eigenvectors

Linear Independence of Eigenvectors

Eigenvectors corresponding to distinct eigenvalues are linearly independent.

Meaning: Different eigenvalues automatically produce independent directions.

Number of Distinct Eigenvalues

If $V$ is finite-dimensional, then

$$\text{number of distinct eigenvalues} \le \dim V.$$

Meaning: An $n$-dimensional space cannot have more than $n$ distinct eigenvalues for one operator.

7) Powers of an Operator

Power of an Operator

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

$$T^0=I$$

and for a positive integer $m$,

$$T^m = \underbrace{T\cdots T}_{m\text{ times}}.$$

Meaning: Powers of an operator mean repeated composition of the operator with itself.

8) Polynomial Applied to an Operator

Polynomial of an Operator

If

$$p(z) = a_0+a_1z+\cdots+a_mz^m,$$

then

$$p(T) = a_0I+a_1T+\cdots+a_mT^m.$$

Meaning: To evaluate a polynomial at an operator, replace the variable by $T$.

Multiplicative Property

For polynomials $p$ and $q$,

$$(pq)(T)=p(T)q(T).$$

Furthermore,

$$p(T)q(T)=q(T)p(T).$$

Meaning: Polynomials of the same operator commute with each other.

9) Invariance from Polynomials of an Operator

Invariant Null Space and Range

For every polynomial $p$,

$$\operatorname{null}p(T)$$

and

$$\operatorname{range}p(T)$$

are invariant under $T$.

Meaning: Polynomial expressions in $T$ naturally produce invariant subspaces.

5B. The Minimal Polynomial

1) Existence of Eigenvalues over $\mathbb{C}$

Existence of Eigenvalues

Every operator on a finite-dimensional nonzero complex vector space has an eigenvalue.

Meaning: Over $\mathbb{C}$, every finite-dimensional nonzero operator has at least one eigenvector direction.

2) Monic Polynomial

Monic Polynomial

A polynomial is monic if its highest-degree coefficient is $1$.

Meaning: A monic polynomial is normalized so that its leading coefficient equals one.

3) Minimal Polynomial

Minimal Polynomial

For finite-dimensional $V$ and $T\in\mathcal{L}(V)$, the minimal polynomial of $T$ is the unique monic polynomial $p$ of smallest degree such that

$$p(T)=0.$$

Meaning: The minimal polynomial is the shortest polynomial relation satisfied by $T$.

Degree Bound

The minimal polynomial satisfies

$$\deg p\le\dim V.$$

Meaning: An operator on an $n$-dimensional space satisfies a polynomial equation of degree at most $n$.

4) Eigenvalues and the Minimal Polynomial

Zeros of the Minimal Polynomial

A scalar $\lambda\in\mathbb{F}$ is an eigenvalue of $T$ if and only if

$$p(\lambda)=0,$$

where $p$ is the minimal polynomial of $T$.

Meaning: The eigenvalues of an operator are exactly the roots of its minimal polynomial.

Complex Factorization

For a complex vector space,

$$p(z) = (z-\lambda_1)\cdots(z-\lambda_m),$$

where the $\lambda_k$ are the eigenvalues of $T$, possibly repeated.

Meaning: Over $\mathbb{C}$, the minimal polynomial factors completely using the eigenvalues.

5) Polynomials that Annihilate an Operator

Polynomial Multiple Criterion

If $p$ is the minimal polynomial of $T$, then

$$q(T)=0$$

if and only if

$$q=ps$$

for some polynomial $s$.

Meaning: Every polynomial relation satisfied by $T$ must contain the minimal polynomial as a factor.

6) Minimal Polynomial of a Restriction

Restriction Property

If $U$ is invariant under $T$, then the minimal polynomial of $T$ is a polynomial multiple of the minimal polynomial of

$$T|_U.$$

Meaning: Restricting an operator to an invariant subspace cannot require a more complicated minimal polynomial.

7) Invertibility and the Minimal Polynomial

Invertibility Criterion

$T$ is not invertible if and only if the constant term of its minimal polynomial is $0$.

Equivalently,

$$T\text{ invertible} \iff p(0)\neq0.$$

Meaning: An operator is noninvertible exactly when $0$ is one of its eigenvalues.

8) Eigenvalues over $\mathbb{R}$

Odd-Dimensional Real Vector Spaces

Every operator on an odd-dimensional real vector space has a real eigenvalue.

Meaning: Although real operators do not always have real eigenvalues, odd dimension guarantees at least one.

5C. Upper-Triangular Matrices

1) Upper-Triangular Matrix

Upper-Triangular Matrix

A square matrix is upper triangular if every entry below the diagonal is $0$.

$$\begin{pmatrix} \lambda_1 & * & \cdots & *\\ 0 & \lambda_2 & \cdots & *\\ \vdots & \ddots & \ddots & \vdots\\ 0 & \cdots & 0 & \lambda_n \end{pmatrix}$$

Meaning: An upper-triangular matrix has zeros everywhere below its diagonal.

2) Invariant-Subspace Criterion

Upper-Triangular Matrix Criterion

For a basis $v_1,\ldots,v_n$, the matrix of $T$ is upper triangular if and only if

$$T v_k \in \operatorname{span}(v_1,\ldots,v_k)$$

for every $k$.

Equivalently,

$$\operatorname{span}(v_1,\ldots,v_k)$$

is invariant under $T$ for every $k$.

Meaning: Upper triangular form corresponds to a nested sequence of invariant subspaces.

3) Polynomial Equation from an Upper-Triangular Matrix

Upper-Triangular Operator Equation

If an upper-triangular matrix of $T$ has diagonal entries

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

then

$$(T-\lambda_1I)\cdots(T-\lambda_nI)=0.$$

Meaning: The diagonal entries produce a polynomial that annihilates the operator.

4) Eigenvalues of an Upper-Triangular Matrix

Diagonal Entries Are Eigenvalues

If the matrix of $T$ is upper triangular, then its eigenvalues are exactly the entries on the diagonal.

Meaning: Eigenvalues can be read directly from the diagonal of an upper-triangular representation.

5) Existence of Upper-Triangular Form

Minimal Polynomial Criterion

$T$ has an upper-triangular matrix with respect to some basis if and only if its minimal polynomial factors completely into degree-one factors:

$$p(z) = (z-\lambda_1)\cdots(z-\lambda_m).$$

Meaning: Upper triangularization is possible exactly when the minimal polynomial splits over the scalar field.

6) Complex Upper-Triangularization

Complex Upper-Triangularization

Every operator on a finite-dimensional complex vector space has an upper-triangular matrix with respect to some basis.

Meaning: Over $\mathbb{C}$, every finite-dimensional operator can be represented in upper-triangular form.

5D. Diagonalizable Operators

1) Diagonal Matrix

Diagonal Matrix

A square matrix is diagonal if every entry outside the diagonal is $0$.

$$\begin{pmatrix} \lambda_1 & 0 & \cdots & 0\\ 0 & \lambda_2 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & \lambda_n \end{pmatrix}$$

Meaning: A diagonal matrix acts independently on each coordinate direction.

2) Diagonalizable Operator

Diagonalizable

An operator is diagonalizable if it has a diagonal matrix with respect to some basis of $V$.

Meaning: A diagonalizable operator becomes simple scaling along suitable basis directions.

3) Eigenspace

Eigenspace

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

$$E(\lambda,T) = \operatorname{null}(T-\lambda I).$$

Equivalently,

$$E(\lambda,T) = \{v\in V\mid Tv=\lambda v\}.$$

Meaning: An eigenspace contains all eigenvectors associated with one eigenvalue, together with the zero vector.

Eigenvalue Criterion

$$\lambda\text{ is an eigenvalue} \iff E(\lambda,T)\neq\{0\}.$$

Meaning: An eigenvalue exists exactly when its eigenspace contains a nonzero vector.

4) Sum of Eigenspaces

Direct Sum of Eigenspaces

If

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

are distinct eigenvalues, then

$$E(\lambda_1,T) \oplus\cdots\oplus E(\lambda_m,T)$$

is a direct sum.

If $V$ is finite-dimensional,

$$\sum_{k=1}^{m}\dim E(\lambda_k,T) \le \dim V.$$

Meaning: Eigenspaces belonging to different eigenvalues contain independent directions.

5) Conditions for Diagonalizability

Diagonalizability Criterion

For finite-dimensional $V$, the following are equivalent:

• $T$ is diagonalizable.
• $V$ has a basis consisting of eigenvectors of $T$.
• $V$ is the direct sum of its eigenspaces.
• The dimensions of all distinct eigenspaces add to $\dim V$.

Thus,

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

Meaning: An operator is diagonalizable exactly when its eigenvectors provide enough independent directions to form a basis.

6) Enough Distinct Eigenvalues

Distinct Eigenvalue Criterion

If

$$T$$

has $\dim V$ distinct eigenvalues, then $T$ is diagonalizable.

Meaning: An $n$-dimensional operator with $n$ distinct eigenvalues automatically has an eigenvector basis.

7) Minimal Polynomial Criterion for Diagonalizability

Diagonalizability and Minimal Polynomial

$T$ is diagonalizable if and only if its minimal polynomial has the form

$$p(z) = (z-\lambda_1)\cdots(z-\lambda_m),$$

where

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

are distinct.

Meaning: An operator is diagonalizable exactly when its minimal polynomial splits into distinct linear factors.

8) Restriction of a Diagonalizable Operator

Restriction Property

If $T$ is diagonalizable and $U$ is invariant under $T$, then

$$T|_U$$

is diagonalizable.

Meaning: Diagonalizability is preserved when restricting to an invariant subspace.

9) Gershgorin Disks

Gershgorin Disk

For a matrix $A$, the $j$th Gershgorin disk is

$$\left\{ z\in\mathbb{F} : |z-A_{j,j}| \le \sum_{\substack{k=1\\k\neq j}}^n |A_{j,k}| \right\}.$$

Meaning: Each disk is centered at a diagonal entry, with radius determined by the other entries in that row.

Gershgorin Disk Theorem

Every eigenvalue of $T$ lies in at least one Gershgorin disk.

Meaning: Gershgorin disks provide regions where the eigenvalues of a matrix must lie.

5E. Commuting Operators

1) Commuting Operators

Commute

Two operators $S$ and $T$ commute if

$$ST=TS.$$

Two matrices $A$ and $B$ commute if

$$AB=BA.$$

Meaning: Commuting operators give the same result regardless of which one is applied first.

2) Operators and Their Matrices

Commutativity and Matrix Representation

With respect to the same basis,

$$ST=TS$$

if and only if

$$\mathcal{M}(S)\mathcal{M}(T) = \mathcal{M}(T)\mathcal{M}(S).$$

Meaning: Commutativity of operators is exactly reflected by commutativity of their matrix representations.

3) Invariance of Eigenspaces

Eigenspace Invariance

If $S$ and $T$ commute, then for every eigenvalue $\lambda$ of $S$,

$$E(\lambda,S)$$

is invariant under $T$.

Meaning: An operator that commutes with $S$ preserves each eigenspace of $S$.

4) Simultaneous Diagonalization

Simultaneous Diagonalizability

Two diagonalizable operators have diagonal matrices with respect to the same basis if and only if they commute.

Meaning: Commuting diagonalizable operators can be diagonalized using one common eigenvector basis.

5) Common Eigenvector

Common Eigenvector Theorem

Every pair of commuting operators on a finite-dimensional nonzero complex vector space has a common eigenvector.

Meaning: Two commuting complex operators always share at least one eigenvector direction.

6) Simultaneous Upper Triangularization

Simultaneous Upper Triangularization

If $S$ and $T$ commute on a finite-dimensional complex vector space, then there exists a basis with respect to which both operators have upper-triangular matrices.

Meaning: Commuting complex operators can be simplified simultaneously using the same basis.

7) Eigenvalues of Sums and Products

Eigenvalues of a Sum

If $S$ and $T$ commute on a finite-dimensional complex vector space, every eigenvalue of

$$S+T$$

is the sum of an eigenvalue of $S$ and an eigenvalue of $T$.

Meaning: Commutativity gives a direct relationship between the eigenvalues of the operators and their sum.

Eigenvalues of a Product

If $S$ and $T$ commute, every eigenvalue of

$$ST$$

is the product of an eigenvalue of $S$ and an eigenvalue of $T$.

Meaning: For commuting operators, eigenvalues of their product come from products of their individual eigenvalues.