Linear Algebra- find an orthogonal matrix with eigenvalue=1 or -1

[RossH]

Homework Statement

I have to find an orthogonal matrix with an eigenvalue that does not equal 1 or -1. That's it. I'm completely stumped.

Homework Equations

An orthogonal matrix is defined as a matrix whose columns are an orthonormal basis, that is they are all orthogonal to each other and each vector has length 1. These matrices have the property that their inverse is the same as their transpose. I don't think their are any other equations.

The Attempt at a Solution

My professor claims that this is possible. So far I thought about a 1x1 matrix, as that is defined as each vector being orthogonal to each other, but the vector only has length 1 if the matrix is [1] or [-1]. And rectangular matrices don't have inverses. I'm stumped.

 

[Mark44]

Homework Statement

I have to find an orthogonal matrix with an eigenvalue that does not equal 1 or -1. That's it. I'm completely stumped.

Homework Equations

An orthogonal matrix is defined as a matrix whose columns are an orthonormal basis, that is they are all orthogonal to each other and each vector has length 1. These matrices have the property that their inverse is the same as their transpose. I don't think their are any other equations.

The Attempt at a Solution

My professor claims that this is possible. So far I thought about a 1x1 matrix, as that is defined as each vector being orthogonal to each other, but the vector only has length 1 if the matrix is [1] or [-1]. And rectangular matrices don't have inverses. I'm stumped.

See if you can cook up a 2x2 matrix that is orthogonal and whose eigenvalues are neither 1 nor -1. Don't limit yourself to real eigenvalues.

 

[Mark44]

BTW, this really should be in the Calculus & Beyond section.

 

[RossH]

See if you can cook up a 2x2 matrix that is orthogonal and whose eigenvalues are neither 1 nor -1. Don't limit yourself to real eigenvalues.

Thanks for the help. I found one:
1/sqrt2 -1/sqrt2
1/sqrt2 1/sqrt2

It's orthogonal and has nonreal eigenvalues. Sorry about putting this post in the wrong forum. I always thought of linear algebra as being a "lower" math than calculus.