Deriving transformation matrix from clues

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SUMMARY

The discussion centers on deriving the transformation matrix T from the equation PTV=R, where P and T are 4x4 matrices and V and R are 4x3 matrices. The user knows P and V, while T is unknown and R has three unknown elements. The user proposes rearranging the equation to T = P'RV', using the left-inverse of P and the pseudoinverse of V, but expresses concern about creating an overdetermined system. The suggestion to analyze the system of linear equations in terms of kernel and rank is presented as a potential approach to solving the problem.

PREREQUISITES
  • Understanding of matrix operations, specifically multiplication and inversion.
  • Familiarity with linear algebra concepts such as transformation matrices and their properties.
  • Knowledge of kernel and rank in the context of linear equations.
  • Experience with pseudoinverses and their applications in solving systems of equations.
NEXT STEPS
  • Study the properties of transformation matrices, focusing on rotation matrices.
  • Learn about the computation and application of left-inverses for non-square matrices.
  • Research the concept of pseudoinverses and their role in solving underdetermined and overdetermined systems.
  • Explore methods for analyzing the kernel and rank of a coefficient matrix in linear algebra.
USEFUL FOR

Mathematicians, engineers, and computer scientists working with linear transformations, as well as students revisiting linear algebra concepts related to matrix equations.

GooseMunch
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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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I would write out the entire system of linear equations, and see what can be said about kernel and rank of its coefficient matrix.
 

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