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Mathematics of Machine Learning
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Definition 25. (Eigenspaces)
Let f : V →V be an arbitrary linear transformation, and λ its eigenvalue. The subspace of eigenvectors defined by
is called the eigenspace of λ.
Eigenspaces play an important role in understanding the structure of linear transformations. First, we note that a linear transformation keeps its eigenspaces invariant. (That is, if x is in the Uλ eigenspace, then f(x) ∈Uλ as well.) This property makes it possible for us to restrict linear transformations to their eigenspaces.
To illustrate the concept of eigenspaces, let’s revisit the already familiar matrix
one more time. Its eigenvalues are λ1 = 3 and λ2 = 1, and by solving the equation (A −λ1I)x = 0, we get that the eigenspace of λ1 is
Similarly, you can check that Uλ2 = {x ∈ℝ2 : x1 = −x2}. (If you go back to Figure 6.2, you can visualize Uλ1 and...