What do eigenvalues and eigenvectors represent intuitively? What is their significance?
In: Mathematics
A matrix represents a transformation. So when you multiply a vector by a matrix, you (usually) get a different vector.
For example, if you have a vector in 3 dimensions (x, y, z), you can multiply it by a 3×3 matrix, where each of the matrixes’ numbers means “which amount of the original vector’s x, y or z goes to the resulting vector’s x, y or z”.
What’s interesting is that every matrix has some vectors that remain “the same” when transformed by the matrix, only scaled up or down. Those are the eigenvectors, and “how much bigger or smaller” they become is their corresponding eigenvalue.
To visualize this: Imagine you have a curved mirror, and when you look at the reflection of a pig, you see a pig squashed to the sides. When you look at the reflection of a cow, you see a stretched up cow. But when you look at your reflection, you see exactly yourself, just two times bigger.
That means you are one of the mirror’s eigenvectors, and your eigenvalue is 2. If you were half your size in the reflection, your eigenvalue would be 0.5.
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