Matrix of orthogonal projection

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Homework Help Overview

The discussion revolves around finding the square of a matrix representing an orthogonal projection, denoted as A^2, through both geometric interpretation and computational methods. The subject area includes linear algebra and matrix theory, particularly focusing on properties of orthogonal projections.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • Participants explore the geometric implications of applying an orthogonal projection twice and question what occurs to a point after multiple projections. There is also an attempt to compute A^2 using the formula for the matrix of orthogonal projection.

Discussion Status

The discussion is active, with participants engaging in both geometric reasoning and computational approaches. Some have raised questions about the outcomes of repeated projections, while others have provided references to external sources for further clarification.

Contextual Notes

There is a noted uncertainty regarding the geometric interpretation of repeated projections, and participants are considering the implications of the definitions and properties of orthogonal projections as they relate to the problem.

morsel
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Homework Statement


Let A be the matrix of an orthogonal projection. Find A^2 in two ways:
a. Geometrically. (consider what happens when you apply an orthogonal projection twice)
b. By computation, using the formula:
matrix of orthogonal projection onto V = QQ^T, where Q = [u1 ... um]


Homework Equations





The Attempt at a Solution


I have no idea how to approach (a).
(b). A^2 = Q Q^T Q Q^T = Q^2 (Q^T)^2

Thanks in advance.
 
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Let's start with (a).
Suppose we have 2 dimensions and A defines an orthogonal projection.
This means that any point v is projected orthogonally on a line.
What happens to this projection if we project it again on the line?
 
Last edited:
Does it go back to the original point?
 
morsel said:
Does it go back to the original point?

We can find on wikipedia a page http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29" :

252px-Orthogonal_projection.svg.png

The transformation P is the orthogonal projection onto the line m.

and:

"In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. "
 
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