Proving Eigenvalues: A Unit Vector Approach for (A - λI)x = b

In summary, the conversation discusses finding the eigenvalue of a unit vector x when given an equation (A - λI)x = b. The solution involves expanding the right hand side and using facts about A, λ, b, and x to show that (A - λI)x = b is true.
  • #1
sana2476
33
0

Homework Statement



Let x be a unit vector. Namely x(Transpose)*x = 1. If (A − Let x be a unit vector. If (A − λI)x = b, then λ is an eigenvalue of A − bx(transpose).


The Attempt at a Solution



I have no idea where to start this proof.
 
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  • #2
Start like this.

[tex]
(A - bx')x = (A - \lambda I + \lambda I - bx')x
[/tex]

Expand the right hand side, and use facts about [tex] A, \lambda, b[/tex] and [tex] x [/tex].
 
Last edited:
  • #3
Ok...so here's what I have so far..I'm not sure if I'm on the right track..

Since x(transpose)*x=1
Therefore, (A-b*x(transpose))*x=λx
So Ax - bx(transpose)x=λx
Ax-b=λx
Then Ax-λx=b
Hence you have (A-λI)x=b

I don't know if I am begging the question by doing it this way though
 
  • #4
No. Look at

[tex] \begin{align*}
(A - bx')x & = (A - \lambda I + \lambda I - bx') x \\
& = (A - \lambda I) x + (\lambda I - bx')x
\end{align*}
[/tex]

What do you know about

[tex]
(A - \lambda I)x
[/tex] ?

use that, together with what you get when you expand

[tex]
(\lambda I - bx') x
[/tex]

(don't forget that [tex] x' x = 1 [/tex])
 

What is an eigenvalue?

An eigenvalue is a scalar value that represents the amount by which a particular vector is stretched or compressed by a linear transformation.

What is the significance of eigenvalues in mathematics and science?

Eigenvalues play a crucial role in various areas of mathematics and science, particularly in linear algebra, quantum mechanics, and differential equations. They allow us to understand the behavior of systems and make predictions about their future states.

How are eigenvalues and eigenvectors related?

An eigenvector is a vector that is unchanged by a linear transformation, except for being multiplied by a scalar value known as the eigenvalue. In other words, the eigenvalue corresponds to the amount of stretching or compression applied to the eigenvector.

Can you explain the concept of diagonalization in relation to eigenvalues?

Diagonalization is the process of finding a diagonal matrix that is similar to a given square matrix. This allows us to simplify complex matrix operations and make use of the easily calculable eigenvalues and eigenvectors.

Why is the proof of eigenvalues important in mathematics and science?

The proof of eigenvalues provides a rigorous and formal justification for the existence and properties of eigenvalues. This allows us to confidently use them in various applications and fields, knowing that they are mathematically sound.

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