Question
For the matrix $P=I-A^{\mathrm{T}}\left(A A^{\mathrm{T}}\right)^{-1} A$, show that if $x$ is in the nullspace of $A$, then $P x=x$. The nullspace stays unchanged under this projection.
Step 1
Step 1: First, we consider the given equation $P x = I x - A^T (A A^T)^{-1} A x$. Show more…
Show all steps
Your feedback will help us improve your experience
Nick Johnson and 65 other educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Let P be an n !! n matrix such that (i) P^2 = P and (ii) P = P^t. Let W be the column space of P. In this exercise, we outline a proof that P = P_W. (a) Prove that, for all X ∈ W, PX = X. [Hint: Since X ∈ W, X = PY for some Y ∈ ℝ^n.] (b) Prove that, for all X ∈ W⊥, PX = 0. [Hint: Show that for all Y ∈ ℝ^n, (PX)^tY = 0 and, hence, (PX)^tPX = 0.] (c) Explain how it follows that P = P_W. [Hint: The matrix of a linear transformation is unique.]
(a) If $P=P^{\mathrm{T}} P$, show that $P$ is a projection matrix. (b) What subspace does the matrix $P=0$ project onto?
Orthogonality
Projections and Least Squares
If $A$ is any matrix, show that the linear transformation $L(\vec{x})=A \vec{x}$ from $\operatorname{im}\left(A^{T}\right)$ to $\operatorname{im}(A)$ is an isomorphism. This provides yet another proof of the formula $\operatorname{rank}(A)=\operatorname{rank}\left(A^{T}\right)$
Orthogonality and Least Squares
Least Squares and Data Fitting
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD