Other
MAT2611 Assignment EIGHT 2023
- Institution
- University Of South Africa (Unisa)
UNISA Linear Algebra MAT2611 Assignment EIGHT of 2023 solutions.
[Show more]
Content preview
MAT2611 ASSIGNMENT 8 2023Problem 27
Let 𝐴 be an 𝑛 × 𝑛 matrix, where 𝑛 ≥ 2, 𝑛 ∈ ℕ
Given that 𝐴 is symmetric and orthogonal.
Since 𝐴 is symmetric then 𝐴𝑇 = 𝐴 (1)
Since 𝐴 is orthogonal then 𝐴𝑇 𝐴 = 𝐼𝑛 (2)
On equation (1) to multiply both sides by 𝐴
𝐴𝑇 = 𝐴
𝐴𝑇 𝐴 = 𝐴𝐴
𝐴𝑇 𝐴 = 𝐴2
𝐴2 = 𝐴𝑇 𝐴
To apply the result on equation (2)
𝐴2 = 𝐼𝑛
Let 𝜆 be an eigenvalue of 𝐴.
Therefore, 𝜆2 is an eigenvalue of 𝐴2
To find the eigenvalues 𝛼 of 𝐴2
𝐴2 𝒗 = 𝛼𝒗 where 𝒗 is an 𝑛 × 1 eigenvector of 𝐴2
Now, 𝐴2 = 𝐼𝑛 , therefore:
𝐼𝑛 𝒗 = 𝛼𝒗
𝒗 = 𝛼𝒗
𝒗 − 𝛼𝒗 = 𝟎
(1 − 𝛼)𝒗 = 𝟎
Eigenvectors may not be zero vectors, therefore:
1−𝛼 =0
𝛼=1
The only possible eigenvalue of 𝐴2 is 1
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through EFT, credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying this summary from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller lyzo2005. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy this summary for R200,00. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
53022 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy summaries for 14 years now