Questions for repetition: projectors and applications
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If a matrix is idempotent, then it can have only two eigenvalues: 0 and 1.
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If a matrix is idempotent, then its image is the set of all vectors that are unchanged by this matrix (such vectors are called fixed points).
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The null space and the image of a projector are orthogonal.
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If
is a projector, then
is too. What is the relationship between the image and null space of
and those of
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Using a basis in a subspace
how do you construct a projector
onto
Prove existence, idempotency and symmetry. Prove that
.
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Prove that if
is the projector from the previous exercise, then
projects onto
Thus, for a given
we have a way to find
Use this fact to prove the theorem on the second orthocomplement.
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Prove the generalized Pythagoras theorem.
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Define the OLS estimator for the linear model and derive it. Why does the condition of linear independence of regressors imply that
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How does the simplified derivation of the OLS estimator look like? Why is it not rigorous?
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Find eigenvalues of a projector. In particular, prove that eigenvectors corresponding to
are orthogonal to eigenvectors corresponding to
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Prove the trace-commuting property.
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What is the relationship between the trace and rank of a projector?
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Using
, find the expressions for the fitted value and residual in the linear model theory.
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Define the OLS estimator of
and prove its unbiasedness. Note the crucial role of the assumption
Without it there is no unbiasedness and Gauss-Markov theorem.
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How do you define a standard normal vector? What are its mean and variance?
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How do you define a general normal vector? Find its mean and variance.
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Find the mean and variance of
Prove that
converges in probability to 1.
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Characterize the distribution of the OLS estimator of
Use it to find its mean and variance.
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