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.
Checking if two matrices are similar
Unlock Audio Lesson
Today, we'll check if the two matrices A and B are similar. Can anyone tell me what it means for two matrices to be similar?
I think matrices A and B are similar if they represent the same linear transformation but in different bases.
Exactly! We can determine this by examining their eigenvalues and eigenvectors. Let's look at matrix A. It has an eigenvalue of 2 with algebraic multiplicity 2. What does that tell us?
It means the eigenvalue 2 is repeated.
Correct! But how many linearly independent eigenvectors does A have? If it has fewer than the algebraic multiplicity, what can we conclude?
It has only one eigenvector, so it's not diagonalizable.
That’s right! Therefore, matrices A and B cannot be similar since B is diagonalizable. Always remember: similarity relies not only on eigenvalues but also on the corresponding eigenvectors.
So A not being diagonalizable means it's not similar to B at all!
Exactly! Great summary. Now, let’s go to our second example about diagonalization.
Diagonalization of a Matrix
Unlock Audio Lesson
For our second example, we have matrix A. Who can remind us what we need to do first in diagonalization?
We need to find the characteristic polynomial!
Correct! What's our characteristic polynomial here?
It's \( \text{det}(A - \lambda I) = (4 - \lambda)(3 - \lambda) \) which gives us eigenvalues 4 and 3.
Good job! Since we have distinct eigenvalues, what does this mean for diagonalization?
It means the matrix can be diagonalized!
Exactly! Now, how do we find the eigenvectors for each eigenvalue?
By solving the equations \( (A - 4I)x = 0 \) and \( (A - 3I)x = 0 \).
Correct! After finding our eigenvectors, we form our change-of-basis matrix P. Can anyone tell me what we do next?
We compute D using \( D = P^{-1}AP \).
Very well! This shows how A is similar to the diagonal matrix D, confirming its diagonalizability.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section presents two key examples explaining how to determine if two matrices are similar by analyzing their eigenvalues and eigenvectors. The first example demonstrates that two matrices are not similar due to lack of diagonalizability, while the second shows how to diagonalize a matrix by finding distinct eigenvalues and constructing the appropriate transformation matrix.
Detailed
Section 31.7: Examples
In this section, we explore two significant examples that highlight the concepts of matrix similarity and diagonalization.
Example 1: Checking Similarity
We analyze the matrices:
A = \( \begin{bmatrix} 2 & 1 \ 0 & 2 \end{bmatrix} \)
B = \( \begin{bmatrix} 2 & 0 \ 0 & 2 \end{bmatrix} \)
We find that matrix A has the eigenvalue \( \lambda = 2 \) with an algebraic multiplicity of 2. However, upon calculating the eigenvectors corresponding to this eigenvalue, we observe that there is only one linearly independent eigenvector, which indicates that A is not diagonalizable. As a result, A is not similar to matrix B, which is diagonalizable and hence represents a different linear transformation.
Example 2: Diagonalization Using Similarity
Next, we consider matrix:
A = \( \begin{bmatrix} 4 & 1 \ 0 & 3 \end{bmatrix} \)
We start by deriving its characteristic equation:
\( \text{det}(A - \lambda I) = (4 - \lambda)(3 - \lambda) \)
The eigenvalues are distinct: \( \lambda_1 = 4 \) and \( \lambda_2 = 3 \). Since A has two distinct eigenvalues, it is diagonalizable. We proceed to find the eigenvectors, form the change-of-basis matrix P from these vectors, and compute the diagonal matrix D via diagonalization. Thus, we conclude that A is similar to the diagonal matrix D, confirming its diagonalizable nature.
These practical examples underscore the relationship between eigenvalues, eigenvectors, and the similarity of matrices, essential concepts in linear algebra, particularly in applications like solving systems of linear equations.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Example 2: Diagonalization Using Similarity
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Example 2: Diagonalization Using Similarity
Let
A = \( \begin{bmatrix} 4 & 1 \ 0 & 3 \end{bmatrix} \)
Find if A is diagonalizable.
Solution:
• Characteristic equation: \( \text{det}(A - \lambda I) = (4 - \lambda)(3 - \lambda) \)
• Eigenvalues: \( \lambda_1 = 4, \lambda_2 = 3 \)
• Since eigenvalues are distinct, A is diagonalizable.
• Find eigenvectors:
– For \( \lambda = 4:\) \( (A - 4I)x = 0 \)
– For \( \lambda = 3:\) \( (A - 3I)x = 0 \)
• Construct matrix P with eigenvectors, compute D = P^{-1}AP
Thus, A is similar to D, confirming diagonalization.
Detailed Explanation
In this example, we are determining if matrix A is diagonalizable. To do this, we first find the characteristic equation by taking the determinant of \( A - \lambda I \). The calculation results in the eigenvalues 4 and 3. Since these are distinct, we know that A can have a complete set of linearly independent eigenvectors, hence A is diagonalizable. Next, we find the eigenvectors for each eigenvalue. By forming a matrix P from these eigenvectors, we can compute the diagonal matrix D. This process of diagonalization confirms that A is indeed similar to a diagonal matrix D, reducing complexity in computations.
Examples & Analogies
Imagine a high school where students are organized by class based on their subjects. If each student (student being an eigenvector) can belong to their distinct classes (eigenvalues), we can arrange them neatly into separate groups without confusion (diagonalization). On the other hand, if there are too many mixed classes with overlapping subjects (like matrix A with insufficient distinct linearly independent eigenvectors), we cannot neatly group them by subjects (indicating non-diagonalizability). Diagonalization here is akin to creating a class schedule that clearly defines who belongs to which subject, making it easy to manage.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Matrix Similarity: Two matrices A and B are similar if they represent the same linear transformation under different bases.
-
Diagonalization: A matrix A is diagonalizable if it can be expressed in the form A = PDP^{-1}, where D is diagonal.
-
Eigenvalues and Eigenvectors: The eigenvalues and corresponding eigenvectors of a matrix are crucial for understanding its properties and performing diagonalization.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Example 1 demonstrates that matrices A and B are not similar as A is not diagonalizable.
-
Example 2 shows that matrix A is diagonalizable due to its distinct eigenvalues, allowing for the formation of a diagonal matrix D.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
Evals might seem like a quest, To find which vectors are the best! If they’re all distinct, you’ve got the key, To turn that matrix into D!
📖 Fascinating Stories
-
Once upon a time, matrices A and B were good friends. They shared the same eigenvalues but not the same eigenvectors. When the time came to change their bases, they could not be similar because A was stuck with just one vector, while B danced freely in diagonal form. They learned that to be similar, they needed the same kind of freedom—plenty of eigenvectors!
🧠 Other Memory Gems
-
For Diagonalization: 'E-V-C' means Eigenvalues first, then find Vectors, then Construct D.
🎯 Super Acronyms
D.O.P.E = Diagonalization Requires One Pair of Eigenvalues for Similarity.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Eigenvalue
Definition:
A scalar value that indicates how much a linear transformation stretches or compresses vectors associated with it.
-
Term: Diagonalizable
Definition:
A matrix is diagonalizable if it can be expressed as a product of matrices such that one is a diagonal matrix.
-
Term: Changeofbasis Matrix
Definition:
An invertible matrix that transforms vectors from one basis to another.