Find the Largest Eigenvalue & Eigenvector of A

In summary, the conversation discusses constructing a matrix A from a vector a, where a is an N by 1 vector and a'a = 5. The conversation also mentions finding the largest eigenvalue of A and the corresponding eigenvector. The speaker suggests generating an example vector and constructing A to solve the problem, but is unsure how to obtain A from a. The confusion is cleared up when it is clarified that aa' is a NxN matrix and a'a is a scalar.
  • #1
A=a.a', where a is an N by 1 vector,a'a=5,and T is transpose.
a)Give the largest eigenvalue of A.
b)what is the corresponding eigenvector?

Please help me to solve the problem.
 
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  • #2
To get started see if you can generate an example vector and construct A and see what the answer is for that case. Then ask what choices you had in your example and whether any of those choices change the problem.
 
  • #3
I'm rather confused, because ## a \cdot a'## is a number not a matrix. Where are you supposed to get A from?
 
  • #4
aa' is a NxN matrix. It's a'a that is a scalar.akanksha331, read your personal messages. This thread is closed.
 

1. What is an eigenvalue and eigenvector?

An eigenvalue is a scalar value that represents the scaling factor of an eigenvector when it is transformed by a matrix. An eigenvector is a non-zero vector that remains in the same direction after it is multiplied by a matrix.

2. Why is finding the largest eigenvalue and eigenvector important?

Finding the largest eigenvalue and eigenvector is important because it can provide insight into the behavior and characteristics of a matrix. It can also be used to solve systems of differential equations and to determine the stability of a system.

3. How do you find the largest eigenvalue and eigenvector of a matrix?

To find the largest eigenvalue and eigenvector of a matrix, you can use the power iteration method. This involves repeatedly multiplying the matrix by a random vector and normalizing the result until it converges to the largest eigenvalue and eigenvector.

4. Can a matrix have more than one largest eigenvalue and eigenvector?

No, a matrix can only have one largest eigenvalue and eigenvector. However, it can have multiple eigenvalues and eigenvectors with the same magnitude, which are known as degenerate eigenvalues and eigenvectors.

5. How are eigenvalues and eigenvectors used in real-world applications?

Eigenvalues and eigenvectors are used in a variety of applications, including image and signal processing, machine learning, and quantum mechanics. They can also be used to solve problems in engineering, physics, and economics.

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