LA-C9

Available

Course 9: Eigenvalues & Eigenvectors

Eigenvalues and eigenvectors reveal the intrinsic structure of linear transformations. They enable diagonalization, simplify matrix computations, and provide deep insights into the behavior of linear systems. This course covers the complete spectral theory of matrices.

15-18 hours Advanced Level 10 Objectives

Learning Objectives

Define eigenvalues and eigenvectors for matrices and linear operators.
Compute eigenvalues using the characteristic polynomial det(A - λI) = 0.
Understand algebraic and geometric multiplicities and their relationship.
Prove that eigenvectors for distinct eigenvalues are linearly independent.
Master diagonalization: when and how to diagonalize matrices.
Understand Jordan normal form for non-diagonalizable matrices.
Apply the Cayley-Hamilton theorem to compute matrix inverses and powers.
Connect eigenvalues to determinant, trace, and invertibility.
Compute matrix powers efficiently using diagonalization.
Understand the geometric meaning of eigenvectors as invariant directions.

Prerequisites

LA-C8: Determinants
Matrix operations and inverses
Linear independence and basis
Polynomial algebra
Complex numbers (for complex eigenvalues)

Historical Context

The concept of eigenvalues emerged in the 18th century through the work of Leonhard Euler and Joseph-Louis Lagrange on differential equations. Augustin-Louis Cauchy (1789–1857) developed the characteristic polynomial. Camille Jordan (1838–1922) introduced the Jordan normal form in 1870, providing a canonical form for all matrices. The Cayley-Hamilton theoremwas stated by Arthur Cayley (1821–1895) and proved by William Rowan Hamilton (1805–1865). Eigenvalue theory is central to quantum mechanics, stability analysis, and many areas of applied mathematics.

1. Eigenvalues and Eigenvectors

An eigenvalue $\lambda$ of a matrix $A$ is a scalar such that $Av = \lambda v$ for some nonzero vector $v$ (the eigenvector). Eigenvectors represent invariant directions under the linear transformation.

Definition 1.1: Eigenvalue and Eigenvector

For an $n \times n$ matrix $A$ , a scalar $\lambda$ is an eigenvalue if there exists a nonzero vector $v$ such that $Av = \lambda v$ . The vector $v$ is an eigenvector corresponding to $\lambda$ .

Definition 1.2: Eigenspace

The eigenspace of $\lambda$ is $E_\lambda = \ker(A - \lambda I) = \{v : Av = \lambda v\}$ .

Theorem 1.1: Independence of Eigenvectors

Eigenvectors corresponding to distinct eigenvalues are linearly independent.

Example 1.1: Finding Eigenvalues

For $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ , eigenvalues are $\lambda = 2, 3$ (diagonal entries of triangular matrix).

Remark 1.1: Geometric Meaning

Eigenvectors define directions that are preserved (possibly scaled) by the transformation. The eigenvalue gives the scaling factor.

2. Characteristic Polynomial

The characteristic polynomial provides an algebraic method to find eigenvalues and encodes fundamental information about the matrix.

Definition 2.1: Characteristic Polynomial

The characteristic polynomial of $A$ is $\chi_A(\lambda) = \det(A - \lambda I)$ .

Theorem 2.1: Eigenvalues as Roots

$\lambda$ is an eigenvalue of $A$ if and only if $\chi_A(\lambda) = 0$ .

Definition 2.2: Multiplicities

For eigenvalue $\lambda_0$ :

Algebraic multiplicity $a$ : exponent of $(\lambda - \lambda_0)$ in $\chi_A(\lambda)$
Geometric multiplicity $g$ : $\dim(E_{\lambda_0})$

Theorem 2.2: Multiplicity Inequality

For any eigenvalue, $1 \leq g \leq a$ .

Theorem 2.3: Vieta's Formulas

For an $n \times n$ matrix $A$ :

Sum of eigenvalues = $\text{tr}(A)$
Product of eigenvalues = $\det(A)$

3. Diagonalization

A matrix is diagonalizable if it can be written as $A = PDP^{-1}$ where $D$ is diagonal. This simplifies many computations, especially matrix powers.

Definition 3.1: Diagonalizable Matrix

An $n \times n$ matrix $A$ is diagonalizable if there exists an invertible matrix $P$ and diagonal matrix $D$ such that $A = PDP^{-1}$ .

Theorem 3.1: Diagonalizability Criterion

$A$ is diagonalizable if and only if for each eigenvalue, geometric multiplicity = algebraic multiplicity.

Corollary 3.1: Distinct Eigenvalues

If $A$ has $n$ distinct eigenvalues, then $A$ is diagonalizable.

Theorem 3.2: Matrix Powers

If $A = PDP^{-1}$ , then $A^k = PD^kP^{-1}$ for any $k$ .

Example 3.1: Diagonalization Algorithm

Step 1: Find eigenvalues via $\det(A - \lambda I) = 0$ . Step 2: For each eigenvalue, find eigenvectors by solving $(A - \lambda I)v = 0$ . Step 3: Form $P$ from eigenvectors, $D$ from eigenvalues. Step 4: Verify $A = PDP^{-1}$ .

4. Jordan Normal Form

When a matrix is not diagonalizable, Jordan normal form provides the best possible canonical form. Every matrix over $\mathbb{C}$ is similar to a Jordan form.

Definition 4.1: Jordan Block

A Jordan block of size $k$ with eigenvalue $\lambda$ is:

J_k(\lambda) = \begin{pmatrix} \lambda & 1 & 0 & \cdots \\ 0 & \lambda & 1 & \cdots \\ \vdots & \ddots & \ddots & \ddots \\ 0 & \cdots & 0 & \lambda \end{pmatrix}

Definition 4.2: Jordan Normal Form

A matrix is in Jordan normal form if it is block-diagonal with Jordan blocks along the diagonal.

Theorem 4.1: Jordan Theorem

Every matrix over $\mathbb{C}$ is similar to a Jordan normal form, unique up to the order of blocks.

Definition 4.3: Generalized Eigenvector

A generalized eigenvector of rank $k$ for eigenvalue $\lambda$ is a vector $v$ such that $(A - \lambda I)^k v = 0$ but $(A - \lambda I)^{k-1} v \neq 0$ .

Remark 4.1: When Jordan Form is Needed

Jordan form is needed when geometric multiplicity < algebraic multiplicity for some eigenvalue. The number of Jordan blocks for $\lambda$ equals the geometric multiplicity.

5. Cayley-Hamilton Theorem

The Cayley-Hamilton theorem states that every matrix satisfies its own characteristic polynomial. This provides powerful methods for computing matrix inverses and powers.

Theorem 5.1: Cayley-Hamilton

If $\chi_A(\lambda) = \det(A - \lambda I)$ is the characteristic polynomial of $A$ , then $\chi_A(A) = 0$ (the zero matrix).

Corollary 5.1: Matrix Inverse via Cayley-Hamilton

If $\chi_A(\lambda) = \lambda^n + c_1\lambda^{n-1} + \cdots + c_n$ , then:

A^{-1} = -\frac{A^{n-1} + c_1 A^{n-2} + \cdots + c_{n-1}I}{c_n}

when $c_n \neq 0$ (i.e., $\det(A) \neq 0$ ).

Definition 5.1: Minimal Polynomial

The minimal polynomial $m_A(\lambda)$ is the monic polynomial of smallest degree such that $m_A(A) = 0$ .

Theorem 5.2: Minimal Polynomial Properties

The minimal polynomial divides the characteristic polynomial and has the same roots (eigenvalues).

Example 5.1: Using Cayley-Hamilton

If $\chi_A(\lambda) = \lambda^2 - 5\lambda + 6$ , then $A^2 - 5A + 6I = 0$ , so $A^2 = 5A - 6I$ . Higher powers can be expressed in terms of $A$ and $I$ .

Course 9 Practice Quiz

Questions

Correct

Accuracy

Av = \lambda v

with

v \neq 0

, then

\lambda

is called:

Easy

Not attempted

The characteristic polynomial of

A

is:

Easy

Not attempted

Sum of eigenvalues (with multiplicity) equals:

Medium

Not attempted

A matrix is diagonalizable iff it has:

Easy

Not attempted

A = PDP^{-1}

with

D

diagonal, then

A^k =

Medium

Not attempted

Cayley-Hamilton states that

\chi_A(A) =

Easy

Not attempted

Geometric multiplicity satisfies:

Hard

Not attempted

Every matrix over

\mathbb{C}

has:

Easy

Not attempted

\lambda = 0

is an eigenvalue:

Medium

Not attempted

Eigenvectors for different eigenvalues are:

Medium

Not attempted

Frequently Asked Questions

What's the geometric meaning of eigenvectors?

An eigenvector's direction is preserved by the linear map. It may be stretched (|λ| > 1), shrunk (|λ| < 1), or flipped (λ < 0), but its direction stays the same. Eigenvectors define 'invariant directions' of the transformation.

When is a matrix diagonalizable?

A matrix is diagonalizable iff for each eigenvalue, the geometric multiplicity equals the algebraic multiplicity. Equivalently, it has n linearly independent eigenvectors. Matrices with n distinct eigenvalues are always diagonalizable.

What is Jordan normal form and when is it needed?

Jordan form is the 'best possible' canonical form for any matrix over ℂ. It's needed when a matrix isn't diagonalizable (geometric < algebraic multiplicity). Every matrix over ℂ is similar to a Jordan form, which is block-diagonal with Jordan blocks.

How does Cayley-Hamilton help compute matrix inverses?

If χ_A(λ) = λⁿ + c₁λⁿ⁻¹ + ... + cₙ, then Aⁿ + c₁Aⁿ⁻¹ + ... + cₙI = 0. Rearranging gives A^{-1} = -(Aⁿ⁻¹ + c₁Aⁿ⁻² + ... + cₙ₋₁I)/cₙ when cₙ ≠ 0.

What's the difference between algebraic and geometric multiplicity?

Algebraic multiplicity = number of times λ appears as root of det(A - λI) = 0. Geometric multiplicity = dim(ker(A - λI)) = number of linearly independent eigenvectors. Always: 1 ≤ geometric ≤ algebraic.