Key Concepts of Diagonalization

Why This Matters

Diagonalization is one of the most powerful techniques you'll encounter in linear algebra because it transforms complex matrix operations into simple arithmetic. When you can express a matrix as $A = PDP^{-1}$, suddenly computing $A^{100}$ becomes trivial, solving systems of differential equations becomes manageable, and understanding how a linear transformation behaves becomes intuitive. You're being tested on your ability to recognize when diagonalization is possible, execute the computational steps correctly, and understand why the process works.

The concepts here—eigenvalues, eigenvectors, characteristic polynomials, and multiplicity conditions—form the backbone of advanced applications in differential equations, quantum mechanics, data science, and stability analysis. Don't just memorize the formula $A = PDP^{-1}$; know what each piece represents, why the conditions for diagonalizability exist, and how to troubleshoot when a matrix fails to diagonalize.

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Foundational Definitions

Before diving into techniques, you need rock-solid definitions. These terms appear constantly in exam questions, so precision matters.

Definition of Diagonalization

• Diagonalization converts a square matrix into diagonal form—specifically, expressing $A = PDP^{-1}$ where $D$ contains eigenvalues on its diagonal
• Matrix $P$ contains eigenvectors as columns—this invertible matrix performs the "change of basis" that reveals the diagonal structure
• Diagonal matrices simplify computations dramatically—powers become $A^k = PD^kP^{-1}$, reducing matrix exponentiation to scalar exponentiation

Eigenvalues and Eigenvectors

• Eigenvalues are scalars $\lambda$ satisfying $Av = \lambda v$—they tell you how much the transformation stretches along special directions
• Eigenvectors represent invariant directions—vectors that only get scaled (not rotated) when the matrix acts on them
• The spectrum (set of all eigenvalues) determines matrix behavior—including stability, convergence, and long-term dynamics

Eigenvalue Equation ($Av = \lambda v$)

• This equation defines the eigenvalue-eigenvector relationship—the matrix $A$ acting on $v$ produces a scalar multiple of $v$
• Eigenvectors are special because they resist rotation—they emerge unchanged in direction, only scaled by $\lambda$
• Rearranging gives $(A - \lambda I)v = 0$—this homogeneous system is how you actually compute eigenvectors

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Finding Eigenvalues

The characteristic polynomial is your primary tool for extracting eigenvalues. Understanding this process is essential for every diagonalization problem.

Characteristic Equation and Polynomial

• The characteristic polynomial comes from $\det(A - \lambda I) = 0$—setting this determinant equal to zero yields the eigenvalues as roots
• Roots of the polynomial are the eigenvalues—for a $2 \times 2$ matrix, you'll typically use the quadratic formula; larger matrices may factor or require numerical methods
• Polynomial degree equals matrix size—an $n \times n$ matrix has a degree-$n$ characteristic polynomial with exactly $n$ eigenvalues (counting multiplicity)

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Multiplicity and Diagonalizability Conditions

This is where students often lose points. The relationship between algebraic and geometric multiplicity determines whether diagonalization succeeds or fails.

Algebraic and Geometric Multiplicity

• Algebraic multiplicity counts repeated roots—how many times $\lambda$ appears as a solution to the characteristic equation
• Geometric multiplicity counts independent eigenvectors—the dimension of the eigenspace $\text{null}(A - \lambda I)$
• Key inequality: geometric ≤ algebraic always holds—diagonalization requires equality for every eigenvalue

Conditions for a Matrix to Be Diagonalizable

• Sufficient eigenvectors must exist—specifically, $n$ linearly independent eigenvectors for an $n \times n$ matrix
• Distinct eigenvalues guarantee diagonalizability—if all $n$ eigenvalues differ, you automatically have $n$ independent eigenvectors
• Repeated eigenvalues require checking geometric multiplicity—this is where diagonalization can fail if the eigenspace is "deficient"

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The Theory Behind Diagonalization

Understanding why diagonalization works helps you apply it correctly and recognize edge cases.

Diagonalization Theorem

• A matrix is diagonalizable iff eigenvectors form a basis—this is the fundamental equivalence you need to internalize
• The theorem connects algebra to geometry—eigenvalues (algebraic) and eigenvectors (geometric) must align properly
• Non-diagonalizable matrices require Jordan form—a more complex canonical form for matrices with insufficient eigenvectors

Similarity Transformation

• Matrices $A$ and $B$ are similar if $A = PBP^{-1}$—they represent the same linear transformation in different bases
• Similar matrices share eigenvalues and characteristic polynomials—these are similarity invariants
• Diagonalization is similarity to a diagonal matrix—the "simplest" representative of the similarity class

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Computational Procedure

Exam problems frequently ask you to execute diagonalization step-by-step. Practice this algorithm until it's automatic.

Steps to Diagonalize a Matrix

• Step 1: Solve $\det(A - \lambda I) = 0$ for eigenvalues—expand the determinant and find all roots of the characteristic polynomial
• Step 2: Find eigenvectors by solving $(A - \lambda I)v = 0$—for each eigenvalue, row-reduce and identify the null space basis
• Step 3: Construct $P$ and $D$—columns of $P$ are eigenvectors, diagonal entries of $D$ are corresponding eigenvalues (in matching order!)

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Real-World Applications

Understanding applications helps you contextualize abstract concepts and prepares you for applied problems.

Applications of Diagonalization

• Matrix powers become trivial—$A^k = PD^kP^{-1}$ reduces to exponentiating diagonal entries, crucial for Markov chains and recurrence relations
• Differential equations decouple—systems $\frac{d\mathbf{x}}{dt} = A\mathbf{x}$ separate into independent equations when $A$ is diagonalized
• Principal Component Analysis (PCA) relies on diagonalization—covariance matrix eigenvectors identify directions of maximum variance in data

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Quick Reference Table

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Self-Check Questions

1. If a $3 \times 3$ matrix has eigenvalues $\lambda = 2, 2, 5$, what additional information do you need to determine if it's diagonalizable?

2. Compare and contrast algebraic multiplicity and geometric multiplicity. Why does their relationship determine diagonalizability?

3. Given $A = PDP^{-1}$, explain why computing $A^{50}$ is easier in this form than working with $A$ directly.

4. Two matrices have the same eigenvalues. Are they necessarily similar? Are they necessarily equal? Justify your answer.

5. A matrix fails to diagonalize because one eigenvalue has geometric multiplicity 1 but algebraic multiplicity 2. What does this tell you about the eigenvector situation, and what alternative canonical form might you use?