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I am struggling to understand, conceptually, what it means for a matrix to have more than one eigen value. I know that eigen values are scalars that correspond to eigen vectors, and that eigen vectors are those that, when put through a certain linear transformation, come out as scalar multiples of themselves.

What I'm actually confounded by is how any matrix can have a determined eigen value at all. That is to say, how can you know which scalar multiples of any vector you're going to get when you put it through your linear transformation? Is it true that one matrix is going to scale any kind of vector in a certain kind(s) of way(s)?

I'd appreciate any help you may be able to provide.

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  • $\begingroup$ had you already seen explicit examples, say in $\Bbb R^2$? $\endgroup$
    – janmarqz
    Nov 19 '15 at 2:12
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An eigenvector for a transformation is just a vector which is scaled by the transformation, but otherwise does not leave its line.

A canonical example is the axis of a rotation in three space. Vectors which do not lie on the same line will typically be turned so that they lie in new lines (but not always).

Why is it so strange to have more than one eigenvalue? Another example is a 180 degree turn around, say, the X axis of three space. That one has two eigenvalues, 1 and -1. The plane normal to the axis of rotation contains lots of eigenvectors for value -1.

If you use the reflection on a plane of three space, you'll have the same eigenvalues, but this time the plane of reflection contains lots of eigenvectors for 1, and the normal vector is an eigenvector for -1.

You can even get $n$ distinct eigenvalues for an n by n matrix if you just pick any matrix that is zero off the main diagonal and you choose the diagonal entries to be distinct.

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