Eigenvalues and Orthogonal Matrices: Proving Properties Without Prefix

In summary, we have proven that if A is an nxn matrix such that (A-I)^{2}=O, then if {\lambda} is an eigenvalue of A, then {\lambda}=1. We have also shown that if A is an orthogonal matrix, then det(A)=+-1. This is because det(A*A^T)=1, which can be rewritten as det(A)^2=1. Therefore, det(A)=+-1, and the solutions are x=+-1.
  • #1
Bertrandkis
25
0
Question 1
Let A be an nxn matrix such that [tex](A-I)^{2}=O[/tex] where [tex]O[/tex] is the zero matrix
Prove that if [tex]{\lambda}[/tex] is an eigen value of A then [tex]{\lambda}=1[/tex]
My attempt
If [tex](A-I)^{2}=O[/tex] then [tex]A=I[/tex] (1)
if [tex]{\lambda}[/tex] is an eigen value of A then [tex]Ax={\lambda}x[/tex] (2)
replace (1) in (2) [tex]Ix={\lambda}x[/tex] , but [tex]Ix=x[/tex] therefore [tex]{\lambda}=1[/tex]

Question 2
If A is an orthogonal Matrix, then prove that det(A)=+-1
My attempt
if A is orthogonal then [tex]AA^{T}=I[/tex] and [tex]A^{-1}=A^{T}[/tex]
therefore [tex]AA^{-1}=I[/tex] and [tex]det(AA^{-1})=1[/tex] . Where does the + - comes from?
 
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  • #2
1) You can't say A=I. It's not necessarily true. You do know (A-I)(A-I)x=0. Now let x be an eigenvector...
2) You've got det(A*A^T)=1. What can you say about the relationship between det(AB) and det(A) and det(B)? How about between det(A) and det(A^T)?
 
  • #3
Dick
1)How do you know that (A-I)(A-I)x=0? Why not [tex](A-I)(A-I)x={\lambda}x[/tex].
If (A-I)(A-I)x=0 and x being the eigen vector, this suggests that [tex]{\lambda}=0[/tex].

2)I know that [tex]det(A^{T})=det(A) [/tex] but how does that prove that
[tex]det(A)=+-1)[/tex] ?
 
  • #4
The problem says (A-I)^2=0. Operating on a vector means (A-I)(A-I)x=0. (A-I)x=Ax-Ix. If [tex]Ax={\lambda}x[/tex] that's [tex](\lambda-1)x[/tex]. Now apply the other (A-I). At the end you have a NUMBER times a nonzero vector equaling zero. So the number is zero.

det(A*A^T)=1=det(A^T)*det(A)=det(A)^2. Solving that equation for det(A) is just like solving x^2=1 for x. What are the solutions?
 
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  • #5
Dick's original point was simply that you cannot, from M2= 0, with M a matrix and 0 the zero matrix, conclude that M= 0 because there are "zero-divisors" in the algebra of matrices. For example,
[tex]M= \left(\begin{array}{cc} 1 & 1 \\ -1 & -1 \end{array}\left)[/tex]
has the property that M2= 0.

Of course, saying M2= 0 means that M2x= 0x= 0 for any vector x. Separate that into M(Mx)= 0 and solve "twice".
 
  • #6
for the first question I came up with this: [tex]0=(A-I)^2=A^2-2A+I.[/tex] so if [tex]Ax=\lambda x,[/tex] for some
vector [tex]x \neq 0,[/tex] then [tex]A^2x=\lambda^2x,[/tex] and [tex]0=(\lambda^2 - 2\lambda + 1)x=(\lambda - 1)^2x.[/tex] thus [tex]\lambda = 1.[/tex]
This looks very right, doesn't it?

For question 2, I figue that x^2=1 yields x=+-1.
Thanks To all.
 

What is an EigenValue?

An EigenValue is a scalar value that represents the amount by which a linear transformation changes a vector. It is an important concept in linear algebra and is used to solve systems of linear equations.

What is the significance of proving the existence of one more EigenValue?

Proving the existence of one more EigenValue can help in understanding the behavior of a linear transformation and its effect on vectors. It can also be used to find solutions to systems of equations that were previously unsolvable.

What are the steps involved in proving the existence of one more EigenValue?

The steps involved may vary depending on the specific problem, but generally, it involves defining the linear transformation, finding the characteristic polynomial, solving for the roots of the polynomial, and proving that one of the roots is an EigenValue.

What are some real-world applications of EigenValues?

EigenValues have various applications in different fields such as physics, engineering, and computer science. They are used in image and signal processing, data compression, and solving differential equations, among others.

Why is it important for scientists to understand EigenValues and their proofs?

EigenValues are a fundamental concept in linear algebra and have numerous applications in various scientific fields. Understanding their proofs can help scientists make accurate predictions and solve complex problems in their respective fields of study.

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