What Does a Determinant of 1 in a Transformation Matrix Signify?

Click For Summary

Homework Help Overview

The discussion revolves around the implications of a determinant value of 1 in the context of transformation matrices, particularly in linear algebra.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the significance of a determinant of 1, with some suggesting it indicates a rotation, while others challenge this notion by providing counterexamples and emphasizing the preservation of volume.

Discussion Status

The conversation includes differing perspectives on the meaning of a determinant of 1, with some participants providing clarifications and counterexamples. There is acknowledgment of the complexity of the topic, but no explicit consensus has been reached.

Contextual Notes

Participants are discussing the properties of transformation matrices under the constraints of linear algebra, with some assumptions about the nature of transformations being questioned.

ns5032
Messages
27
Reaction score
0
Does it mean anything in particular about the transformation if the determinant of a transformation matrix is 1?
 
Physics news on Phys.org
Yes, it does. That means the transformation does not change the length of a vector nor does it reverse the direction. It is, basically, a "rotation".
 
det=1 is not sufficient to show a transformation is a rotation, though the converse is true. Consider a matrix like [[1/2,0],[0,2]]. What is true is that the transformation doesn't change the volume of a region.
 
Thanks, Dick. You are, of course, right.
 

Similar threads

What does a row matrix represent geometrically?
  • · Replies 8 ·
Replies
8
Views
1K
Linear Algebra Determinant proof
  • · Replies 4 ·
Replies
4
Views
1K
How to find the determinant of this matrix?
  • · Replies 3 ·
Replies
3
Views
1K
Diagonalizing a matrix given the eigenvalues
  • · Replies 8 ·
Replies
8
Views
2K
Find Matrix P and the Diagonal Matrix D
  • · Replies 2 ·
Replies
2
Views
1K
Vector Subspaces: Determining U as a Subspace of M4x4 Matrices
  • · Replies 8 ·
Replies
8
Views
2K
Solving a separable matrix ODE.
  • · Replies 3 ·
Replies
3
Views
2K
Show that this column matrix is not a vector
  • · Replies 2 ·
Replies
2
Views
1K
Why is my calculation for the determinant of a matrix incorrect?
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Similarity transformation changing the determinant to 1
  • · Replies 20 ·
Replies
20
Views
3K