Eigenvalue for Orthogonal Matrix

In summary, the conversation discusses the proof that an eigenvector belonging to an eigenvalue of an orthogonal matrix Q is also an eigenvector of Q^T. The conversation includes the relevant equations and a question about the starting point for the proof. The solution is found by using the definition of an orthogonal matrix.
  • #1
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Homework Statement



Let Q be an orthogonal matrix with an eigenvalue [itex]λ_{1}[/itex] = 1 and let x be an eigenvector belonging to [itex]λ_{1}[/itex]. Show that x is also an eigenvector of [itex]Q^{T}[/itex].

Homework Equations



Qx = λx where x [itex]\neq[/itex] 0

The Attempt at a Solution



[itex]Qx_{1} = x_{1}[/itex] for some vector [itex]x_{1}[/itex]

[itex](Qx_{1})^{T} = x_{1}^{T}Q^{T}[/itex]


I'm kind of stuck with how to start this problem, as I'm not sure what I've done is even starting down the right path. Can anyone give me a nudge in the right direction?
 
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  • #2
What's (Q^T)(Q) if Q is real orthogonal?
 
  • #3
Thank you. I didn't realize what an orthogonal matrix was (yikes!). Once I did the proof fell right out of the definition.
 

Related to Eigenvalue for Orthogonal Matrix

1. What is an orthogonal matrix?

An orthogonal matrix is a square matrix where the rows and columns are orthonormal, meaning they are all perpendicular to each other and have a magnitude of 1. In other words, an orthogonal matrix is a special type of matrix where the transpose is equal to the inverse.

2. What is the significance of eigenvalues for orthogonal matrices?

Eigenvalues for orthogonal matrices have a special property where they are all either 1 or -1. This means that when an orthogonal matrix is multiplied by its transpose, the resulting matrix will have eigenvalues equal to 1. This property is important in many applications, including solving systems of linear equations and in principal component analysis.

3. How are eigenvalues calculated for orthogonal matrices?

Since the eigenvalues for orthogonal matrices are all 1 or -1, they can be easily calculated by finding the determinant of the matrix. If the determinant is positive, then the eigenvalues will be 1, and if the determinant is negative, the eigenvalues will be -1.

4. What is the relationship between eigenvalues and eigenvectors for orthogonal matrices?

For orthogonal matrices, the eigenvectors corresponding to the eigenvalue of 1 are called right eigenvectors, while the eigenvectors corresponding to the eigenvalue of -1 are called left eigenvectors. These eigenvectors are orthogonal to each other and have a magnitude of 1.

5. What are some real-world applications of orthogonal matrices and eigenvalues?

Orthogonal matrices and eigenvalues are used in a variety of fields, including computer graphics, engineering, and statistics. Some examples of real-world applications include image compression, data compression, and signal processing.

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