What are eigenvectors and eigenvalues?
— Eigenvectors are special vectors that remain on their own line during a transformation, and eigenvalues are the factors by which they are stretched or squished.
How can eigenvectors and eigenvalues simplify matrix operations?
— Eigenvectors and eigenvalues can be used to understand linear transformations and are often a better way to get at the heart of what the transformation does than reading off the columns of the matrix.
What is the significance of λ in matrix-vector multiplication?
— In matrix-vector multiplication, λ represents the corresponding eigenvalue and is used in the transformation associated with the matrix.
How can diagonal matrices make operations easier?
— Diagonal matrices are easier to work with as they can be multiplied by themselves multiple times to scale each basis vector by the corresponding eigenvalue.
Why is changing to an eigenbasis beneficial?
— Changing to an eigenbasis and computing the power of a transformation in that system makes it easier to compute the power in the original coordinate system.
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