Orthogonal projection matrices

[MathewsMD]

I've attached the question to this post. The answer is false but why is it not considered the orthogonal projection?

$A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$
$B = \begin{bmatrix} x \\ y \end{bmatrix}$
$AB = \begin{bmatrix} y \\ 0 \end{bmatrix}$

Is AB not providing the orthogonal projection? Is my matrix multiplication incorrect?

 

[Stephen Tashi]

The question was about orthogonal projection "onto the x-axis", so (x,y) should project to (x,0).

 

[MathewsMD]

The question was about orthogonal projection "onto the x-axis", so (x,y) should project to (x,0).

I was interpreting it as the projection of the orthogonal component (i.e. y) to the x-axis. I understand what you're saying, thank you for the clarification, but why exactly is it called the orthogonal projection then? Why not simply the projection to the x-axis?

 

[Stephen Tashi]

why exactly is it called the orthogonal projection then? Why not simply the projection to the x-axis?

A "projection" is defined to be a transformation that produces the same answer when applied twice as when applied once. For example, we can define a projection process on (x,y) to be drawing a line through (x,y) that makes a 120 degree angle with the x-axis and mapping (x,y) to the point of intersection between that line and the x-axis. Applying the process twice means applying it to the point (x,y) and then to the point where (x,y) was mapped.

In an "orthogonal projection onto the x-axis, " the direction of the line we draw would be perpendicular to the x-axis. Do your text materials give a precise definition for "orthogonal projection"? Perhaps they define it in terms of an inner product.

(Don't confuse "orthogonal projections" with "orthogonal matrices", which is another topic you will encounter in linear algebra.)

 

[blue_raver22]

The statement that AB is not considered an orthogonal projection is false. The matrix AB does indeed represent an orthogonal projection onto the y-axis. This can be seen by considering the definition of an orthogonal projection matrix, which is a square matrix that is equal to its own transpose and whose square is itself. In this case, AB is a 2x2 matrix that is equal to its own transpose and whose square is itself, satisfying the criteria for an orthogonal projection matrix.

Your matrix multiplication is correct. AB does provide the orthogonal projection onto the y-axis, as it maps any vector (x,y) to the vector (y,0), which lies on the y-axis. Therefore, AB is an orthogonal projection matrix and your understanding of orthogonal projection is correct.