Deriving transformation matrix from clues

In summary, The conversation discusses the problem of finding a transformation matrix T, using known square matrices P and V, and a partially known non-square matrix R. The speaker is unsure of how to approach the problem, but suggests rearranging the equation to T = P'RV', where P' and V' are the inverses of P and V respectively. However, this may lead to an overdetermined system and the speaker is seeking guidance on how to proceed.
  • #1
GooseMunch
1
0
I have PTV=R where P and T are square matrices (4x4) and V and R are non-square (4x3).
P and V are known, T is unknown, and R is partially known (3 unknown elements).
Seems impossible, but T is a transformation matrix (ie upperleft 3x3 is a rotation matrix) which gives me additional clues.

I'm trying to find T, but I can't figure out how to approach this problem. It's been almost 10 years since I took linear algebra course and I feel very lost.
Seems like I can just rearrange it like so:
T = P'RV' where P' and V' are inverses of P and V respectively but V is non-square so I can only get left-inverse. I don't think pseudoinverse of V is the answer because I'm afraid I'll end up with overdetermined system and the whole thing becomes unsolvable.

Can anyone nudge me in the right direction? Thanks.
 
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  • #2
I would write out the entire system of linear equations, and see what can be said about kernel and rank of its coefficient matrix.
 

FAQ: Deriving transformation matrix from clues

How do you derive a transformation matrix from clues?

To derive a transformation matrix from clues, you need to gather information about the original and transformed coordinates of points on a graph. Then, you can use this information to create a system of equations and solve for the values of the transformation matrix.

What are some common clues used to derive a transformation matrix?

Some common clues used to derive a transformation matrix include the coordinates of a point before and after a transformation, the angle of rotation, and the scale factor. Other clues may include the direction and magnitude of translation or the coordinates of multiple points on a graph.

Is it possible to derive a transformation matrix with just one clue?

No, it is not possible to derive a transformation matrix with just one clue. You need at least two clues to create a system of equations and solve for the values of the transformation matrix. However, the more clues you have, the more accurate and precise the transformation matrix will be.

What is the purpose of deriving a transformation matrix?

The purpose of deriving a transformation matrix is to accurately describe a transformation that occurs on a graph. This matrix can then be used to apply the same transformation to other points on the graph, making it a useful tool in geometry, computer graphics, and other fields.

Are there any shortcuts or tricks for deriving a transformation matrix?

While there are no shortcuts or tricks for deriving a transformation matrix, there are some strategies that can make the process easier. These include using a matrix calculator, organizing the clues in a table, and checking the matrix by applying it to other points on the graph.

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