Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Algebra → Linear Algebra ↓


Diagonalization


Diagonalization is a concept in linear algebra that involves transforming a square matrix into diagonal form. In simple terms, diagonalization is the process of finding a diagonal matrix that is similar to a given square matrix. This process is incredibly useful because diagonal matrices are the simplest form of matrix and are quite easy to work with, especially for calculations involving powers of matrices and solving differential equations.

Before getting into diagonalization, it is important to familiarize yourself with some basic concepts like matrix, eigenvectors, and eigenvalues, as they play a vital role in diagonalization.

Understanding matrices

A matrix is a rectangular array of numbers that can be used to represent linear transformations and systems of linear equations. A matrix with m rows and n columns is called mxn matrix:

A = [a 11 a 12 a 13
     a 21 a 22 a 23
     a 31 a 32 a 33 ]

In this example, matrix A is a 3 x 3 matrix.

Eigenvalues and eigenvectors

One of the key concepts of diagonalization is eigenvalue and eigenvector. Suppose A is a square matrix. A non-zero vector v is called an eigenvector of A if there exists a scalar λ (lambda) such that:

 a * v = λ * v

In this equation, λ is called the eigenvalue associated with the eigenvector v. Essentially, when a matrix acts on one of its eigenvectors, it simply stretches or contracts it without changing its direction.

Diagonalization process

To determine if a matrix A is diagonalizable, we need to see if it can be expressed as:

 A = P * D * P -1

Here, matrix D is a diagonal matrix and matrix P is an invertible matrix whose columns are the eigenvectors of A. The diagonal elements of D are the eigenvalues of A.

Let's outline the steps to diagonalize a matrix:

Step 1: Find the eigenvalues

The eigenvalues of the matrix A are found by solving the characteristic polynomial:

 det(A - λI) = 0

Here, I is the identity matrix of the same size as A, and λ denotes the eigenvalues.

Step 2: Find the eigenvectors

Determine the eigenvectors by solving for each eigenvalue:

 (A – λI)v = 0

The solutions v are the eigenvectors associated with the eigenvalue λ.

Step 3: Create matrices P and D

Construct the matrix P using the eigenvectors as columns. The diagonal matrix D will have the eigenvalues on its diagonal.

Step 4: Verify diagonalization

Check that:

 A = P * D * P -1

If this equation is true, then the matrix A is diagonalizable.

Example

Consider the matrix:

 A = [4 1
     2 3]

Step 1: Find the eigenvalues

The characteristic polynomial is:

 det(A - λI) = det([4-λ 1
                          2 3 - λ])

Calculate the determinant:

 (4 – λ)(3 – λ) – (2)(1) = λ 2 – 7λ + 10

Set this equation to zero and solve for λ:

 λ 2 - 7λ + 10 = 0

Let's factor this in:

 (λ – 5)(λ – 2) = 0

Thus, λ = 5 and λ = 2 are the eigenvalues.

Step 2: Find the eigenvectors

For λ = 5, we solve:

 (A – 5I)v = 0
 [4-5 1 
   2 3-5]v = [-1 1 
              2 -2]

Solving the system, we find the eigenvectors:

 v 1 = [1 
                   1]

For λ = 2, we solve:

 (A – 2I)v = 0
 [4-2 1 
   2 3-2]v = [2 1 
              2 1]

Solving the system, we find the eigenvectors:

 v 2 = [1 
                   -2]

Step 3: Create matrices P and D

Create the matrix P and the diagonal matrix D using the eigenvectors:

 P = [1 1 
      1 -2]
 D = [5 0 
      0 2]

Step 4: Verify diagonalization

Now verify that multiplying P, D, and P -1 gives the original matrix A:

 P * D * P -1 = [1 1 
                           1 -2] * [5 0 
                                    0 2] * [1 1 
                                            1 -2]

This confirms that the matrix A is diagonalizable.

Why diagonalization is useful

Diagonalization greatly simplifies many linear algebra calculations. For example, when a matrix is raised to a large power, diagonalization allows the process to be feasible and simple. Let's look at an example of raising our matrix A to a power of 3:

Example: computational efficiency

Instead of multiplying A by itself several times, A 3 is simplified using diagonalization:

 A 3 = (P * D * P -1) 3 = P * D 3 * P -1

Calculate D 3 by raising each diagonal element of D to the power of 3:

 D 3 = [5 3 0 
                   0 2 3]
                 = [125 0 
                    0 8]

Now calculate:

 A 3 = P * D 3 * P -1

Computing this multiplication is generally straightforward. Diagonalization provides practical computational advantages in a variety of applications, such as transforming systems of linear equations, efficiently computing matrix exponents, simplifying linear transformations, and many more.

Limitations and conditions

While diagonalization greatly simplifies calculations, not every matrix can be diagonalized. Diagonalization is only possible if the matrix has enough linearly independent eigenvectors to form the matrix P. Specifically, an nxn matrix can only be diagonalized if it has n linearly independent eigenvectors. Matrices that do not have these eigenvectors or have repeated eigenvalues, without corresponding eigenvectors that can form the required n linearly independent sets, are not diagonalizable. These matrices are called defective matrices.

Understanding when diagonalization is possible is important in determining how to effectively simplify complex matrix operations. This process also extends beyond pure mathematics into applied fields such as physics, engineering, and computer science, where linear transformations are important.


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