Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Characteristic Equation
Unlock Audio Lesson
To begin finding the eigenvalues for our matrix A, we first need to establish the characteristic equation. Can anyone remind us how we compute this?
Is it through the determinant of (A - λI)?
Exactly! We take our matrix A, subtract λ times the identity matrix I, and compute the determinant. For our example, A is [[4, 1], [0, 4]].
What do we get when we set that determinant to zero?
We solve det(A - λI) = 0. This leads us to the eigenvalues. Let’s calculate that.
Eigenvalues and Eigenvectors
Unlock Audio Lesson
Now that we discovered λ = 4 as our eigenvalue, what does that imply about our eigenspace?
We have to find the eigenspace corresponding to that eigenvalue?
Correct! We solve (A - 4I)v = 0. What do we need to keep in mind when solving this equation?
We're looking for a basis of eigenvectors that satisfies this equation.
Exactly, and we’ll analyze the dimension of this space next!
Eigenspace and Basis
Unlock Audio Lesson
After solving (A - 4I)v = 0, we found that the eigenspace is spanned by the vector [1, 0]. What’s the significance of finding only one linearly independent eigenvector?
It means the geometric multiplicity is less than the algebraic multiplicity?
Precisely! Since GM = 1 and AM = 2, what does that tell us about the matrix?
It is not diagonalizable.
Application of Non-Diagonalizable Matrices
Unlock Audio Lesson
So we learned this matrix isn't diagonalizable. Why might that cause issues in engineering applications?
It could complicate the analysis of structural modes or vibrations, right?
Exactly! In engineering, dealing with non-diagonalizable matrices means we have to be careful about assumptions we make in modal analysis.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section outlines how to solve the characteristic equation of a matrix to find its eigenvalues, followed by determining the corresponding eigenspace and basis of eigenvectors. This specific example reveals an instance of a repeated eigenvalue and its implications in linear algebra.
Detailed
Detailed Summary
This section demonstrates the process of finding eigenvalues and eigenvectors of the matrix A = [[4, 1], [0, 4]]. The first step involves solving the characteristic equation defined by the determinant of (A - λI). The solution reveals that λ = 4 is a repeated eigenvalue with algebraic multiplicity (AM) of 2. Next, the section delves into the eigenspace corresponding to this eigenvalue by solving the equation (A - 4I)v = 0. The result indicates that there is only one linearly independent eigenvector, leading to a geometric multiplicity (GM) of 1, which is less than the algebraic multiplicity. Consequently, the matrix is determined not to be diagonalizable. The section highlights the significance of these concepts in the broader context of linear algebra applications, particularly in engineering.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Step 1: Characteristic Equation
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Let:
A=
[4 1]
[0 4]
Step 1: Characteristic Equation
det(A−λI)=det
[4−λ 1 ]
[0 4−λ]
So, λ=4 is a repeated eigenvalue (AM = 2).
Detailed Explanation
In this step, we need to find the characteristic equation of the matrix A. This is done by calculating the determinant of the matrix (A - λI), where λ represents the eigenvalue and I is the identity matrix. The result shows that there is a repeated eigenvalue λ = 4, meaning the algebraic multiplicity (the number of times the eigenvalue appears) is 2.
Examples & Analogies
Think about how many times your favorite item appears in a store. If there are two of the same product on the shelf, then in terms of eigenvalues, we can say that there is a repeated value. Here, our 'product' is the eigenvalue '4' that appears more than once.
Step 2: Eigenspace
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Step 2: Eigenspace
Solve (A−4I)v=0:
[0 1][x] = [0]
[0 0] [y] = [0]
So, y=0
So, x∈R, and the eigenspace is:
E = Span{[1 0]}
Only one linearly independent eigenvector ⇒ GM = 1 < AM = 2 ⇒ Matrix is not diagonalizable.
Detailed Explanation
Here, we are solving the equation derived from the eigenvalue we found earlier (λ = 4). By substituting λ into (A - λI) and solving for the vector v, we observe that y must equal 0. Consequently, we conclude that x can be any real number (x ∈ R). This gives us the eigenspace E, which spans in the direction of the vector [1, 0]. Since we found only one linearly independent eigenvector, the geometric multiplicity (GM) is 1, which is less than the algebraic multiplicity (AM) of 2. Therefore, this matrix cannot be diagonalized.
Examples & Analogies
Imagine you have a group of people where only one person stands out because they are different. In this context, that 'one person' corresponds to the one linearly independent eigenvector we found, while the others who align with this person's direction represent higher multiplicities that don't offer any distinctiveness.
Basis of Eigenvectors for λ=4
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Basis of eigenvectors for λ=4:
{[1]}
0
Detailed Explanation
This part states that the only basis of eigenvectors corresponding to the repeated eigenvalue λ=4 consists of the vector [1, 0]. This means any vector in the eigenspace can be expressed as a scalar multiple of this basis vector.
Examples & Analogies
It's like having a single recipe that can be altered by changing just the quantity of one ingredient. Here, our basis vector [1, 0] is that primary recipe, and any vector in the eigenspace is simply a variation of it based on how much of it we 'use'.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Characteristic Equation: A mathematical equation used to find eigenvalues.
-
Eigenspace: The vector space corresponding to an eigenvalue, consisting of all its eigenvectors.
-
Basis of Eigenvectors: Set of linearly independent eigenvectors that span the eigenspace.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Example 1: For the matrix A = [[4, 1], [0, 4]], we find λ = 4 is a repeated eigenvalue leading to a basis of eigenvectors { [1, 0] }.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
Eigenvalues rise, vectors they prize, form a space where solutions lie.
📖 Fascinating Stories
-
Imagine a treasure map. The eigenvalues are clues leading to the treasure (eigenvectors) hidden in the eigenspace.
🧠 Other Memory Gems
-
AM - A Measure, GM - A Geometry: Remember 'A Measure' for Algebraic and 'Geometry' for Geometric.
🎯 Super Acronyms
EG
- Eigenvalues Generate - think of how eigenvalues lead you to eigenvectors.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Eigenvalue
Definition:
A scalar λ such that Av = λv for a non-zero vector v.
-
Term: Eigenvector
Definition:
A non-zero vector v that satisfies Av = λv.
-
Term: Eigenspace
Definition:
The set of all eigenvectors corresponding to a particular eigenvalue, including the zero vector.
-
Term: Algebraic Multiplicity (AM)
Definition:
The number of times an eigenvalue appears as a root of the characteristic polynomial.
-
Term: Geometric Multiplicity (GM)
Definition:
The dimension of the eigenspace corresponding to an eigenvalue.