Question

(a) Let A be a 4x4 matrix with det(A) = 0. If 1, 3 and 5 are eigenvalues of A, give one reason why A is diagonalizable. (b) If A is a 4 !! 4 matrix with three distinct eigenvalues ?1, ?2, ?3 (all real numbers) and if the dimension of the eigenspace E?1 is 2, prove that A is diagonalizable. (c) Suppose that A is a 3 !! 3 diagonalizable matrix with a basis of eigenvectors of ?3 given by {u1, u2, u3} with corresponding eigenvalues ?1, ?2, ?3. Suppose that the matrix B is also a 3 !! 3 diagonalizable matrix with the same eigenvectors although with possibly different eigenvalues ?1, ?2, ?3. Prove that A + B is also a diagonalizable matrix.

          (a) Let A be a 4x4 matrix with det(A) = 0. If 1, 3 and 5 are eigenvalues of A, give one reason why A is diagonalizable.

(b) If A is a 4 !! 4 matrix with three distinct eigenvalues ?1, ?2, ?3 (all real numbers) and if the dimension of the eigenspace E?1 is 2, prove that A is diagonalizable.

(c) Suppose that A is a 3 !! 3 diagonalizable matrix with a basis of eigenvectors of ?3 given by {u1, u2, u3} with corresponding eigenvalues ?1, ?2, ?3. Suppose that the matrix B is also a 3 !! 3 diagonalizable matrix with the same eigenvectors although with possibly different eigenvalues ?1, ?2, ?3. Prove that A + B is also a diagonalizable matrix.

Show more…

(a) Let A be a 4x4 matrix with det(A) = 0. If 1, 3 and 5 are eigenvalues of A, give one reason why A is diagonalizable.

(b) If A is a 4 !! 4 matrix with three distinct eigenvalues ?1, ?2, ?3 (all real numbers) and if the dimension of the eigenspace E?1 is 2, prove that A is diagonalizable.

(c) Suppose that A is a 3 !! 3 diagonalizable matrix with a basis of eigenvectors of ?3 given by u1, u2, u3 with corresponding eigenvalues ?1, ?2, ?3. Suppose that the matrix B is also a 3 !! 3 diagonalizable matrix with the same eigenvectors although with possibly different eigenvalues ?1, ?2, ?3. Prove that A + B is also a diagonalizable matrix.

Added by Salvador V.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

08/31/2023

Step 1

Now, we know that A has eigenvalues 1, 3, and 5. Since A is a 4x4 matrix, it can have at most 4 distinct eigenvalues. Show more…

Show all steps

lock

Thanks for your feedback!

(a) Let A be a 4x4 matrix with det(A) = 0. If 1, 3 and 5 are eigenvalues of A, give one reason why A is diagonalizable. (b) If A is a 4 !! 4 matrix with three distinct eigenvalues λ1, λ2, λ3 (all real numbers) and if the dimension of the eigenspace Eλ1 is 2, prove that A is diagonalizable. (c) Suppose that A is a 3 !! 3 diagonalizable matrix with a basis of eigenvectors of ℝ3 given by {u1, u2, u3} with corresponding eigenvalues λ1, λ2, λ3. Suppose that the matrix B is also a 3 !! 3 diagonalizable matrix with the same eigenvectors although with possibly different eigenvalues ℓ1, ℓ2, ℓ3. Prove that A + B is also a diagonalizable matrix.

Play audio

Feedback

Powered by NumerAI

Sri K and 97 other subject Calculus 3 educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Recommended Videos

Recommended Textbooks

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later