Building Projection Matrices from \delta_{ij} and M_{ij}

In summary, the conversation is about trying to find projection matrices A_{ij} and B_{ij} that satisfy orthonormality conditions. However, this seems impossible due to the first condition showing that the determinants of the matrices can't both be non-zero. The existence of the real non-invertible symmetric matrix M further complicates the possibility of constructing these matrices.
  • #1
TriTertButoxy
194
0
Out of the unit matrix and a real non-invertible symmetric matrix of the same size,

[tex]\delta_{ij}[/tex] and [tex]M_{ij}[/tex]​

I need to build a set of projection matrices, [itex]A_{ij}[/itex] and [itex]B_{ij}[/itex] which satisfy orthonormality:

[tex]A_{ij} B_{jk}=0,[/tex] and [tex]A_{ij} A_{jk}=B_{ij} B_{jk}=\delta_{ik}[/tex]​

Is this possible? or should I give up trying to find such matrices?
 
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  • #2
Where does the matrix M come in?

I don't think what you're requesting is possible. Just writing in terms of matrices, you want AB = 0 and A2 = B2 = I. But the first condition shows that det(A)det(B) = 0, so det(A) = 0 or det(B) = 0. If det(A) = 0, then det(A2) = 0, making A2 = I impossible.
 
  • #3
Good point; there are no such matrices I can construct. Thanks.
 

Related to Building Projection Matrices from \delta_{ij} and M_{ij}

1. What is a projection matrix?

A projection matrix is a square matrix that maps vectors onto a subspace by projecting them onto a lower-dimensional space. It is commonly used in linear algebra and computer graphics to transform and manipulate objects.

2. How is a projection matrix constructed?

A projection matrix can be constructed using the Kronecker delta function (\delta_{ij}) and a matrix M_{ij}. The Kronecker delta function is used to create an identity matrix, while the matrix M_{ij} determines the properties of the projection, such as the direction and size.

3. What are the properties of a projection matrix?

A projection matrix has the following properties:

  • It is a square matrix (n x n).
  • It is symmetric and idempotent (P^2 = P).
  • Its eigenvalues are either 0 or 1.
  • Its nullspace is the subspace onto which vectors are projected.
  • Its range is the subspace onto which vectors are projected.

4. How is a projection matrix used in computer graphics?

In computer graphics, a projection matrix is used to transform 3D objects into a 2D image on a screen. It is often used in conjunction with a viewing matrix to determine the perspective and position of the object in the scene.

5. Can a projection matrix be used for non-linear projections?

No, a projection matrix can only be used for linear projections. Non-linear projections require more complex transformations, such as using a non-linear function or a non-square matrix.

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