Proving Trace Independence in Linear Algebra Homework

Click For Summary
SUMMARY

The discussion focuses on proving that the trace of a matrix, defined as Trace(A) = Ʃ(i=1..n) (ei|A|ei), is independent of the chosen orthonormal basis. Participants highlight that the trace is directly related to the eigenvalues of the matrix, which remain constant regardless of the basis. The challenge lies in demonstrating the relationship between a matrix A and its representation in different bases, particularly when A is diagonalizable and the ei are eigenvectors.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly matrix representation.
  • Familiarity with eigenvalues and eigenvectors.
  • Knowledge of diagonalizable matrices.
  • Basic grasp of orthonormal bases in vector spaces.
NEXT STEPS
  • Study the properties of diagonalizable matrices in linear algebra.
  • Learn how to transform matrices between different bases.
  • Explore the relationship between eigenvalues and matrix trace.
  • Investigate the implications of basis independence in linear transformations.
USEFUL FOR

Students of linear algebra, educators teaching matrix theory, and anyone interested in the foundational properties of matrix operations and their implications in various mathematical contexts.

zheng89120
Messages
139
Reaction score
0

Homework Statement



let:
Trace(A) = Ʃ(i=1..n) (ei|A|ei)

Show that trace is independent of the orthonormal basis chosen.

Homework Equations



linear algebra

The Attempt at a Solution



trace is related to the eigenvalues, which are constant?
 
Physics news on Phys.org
zheng89120 said:

Homework Statement



let:
Trace(A) = Ʃ(i=1..n) (ei|A|ei)

Show that trace is independent of the orthonormal basis chosen.

Homework Equations



linear algebra

The Attempt at a Solution



trace is related to the eigenvalues, which are constant?

IF A a diagonalizable and IF the ei happen to be eigenvectors, then sure, it's easy to see the trace is the sum of the eigenvalues. But how does that help you show it's basis independent? How do you write the relation between A and A written in a different basis?
 

Similar threads

Introduction to Linear Algebra: Solving Real-World Problems
  • · Replies 10 ·
Replies
10
Views
2K
Isomorphisms preserve linear independence
  • · Replies 2 ·
Replies
2
Views
2K
Conflicting definitions of linear independence
  • · Replies 30 ·
2
Replies
30
Views
2K
Dimension statement about (finite-dimensional) subspaces
  • · Replies 15 ·
Replies
15
Views
2K
Null space of dual map of T = annihilator of range T
  • · Replies 1 ·
Replies
1
Views
3K
How would you explain to a dummy how a diff. eq. is linear?
  • · Replies 6 ·
Replies
6
Views
2K
Determinant of exponential matrix
  • · Replies 6 ·
Replies
6
Views
3K
Dimension of orthogonal subspaces sum
  • · Replies 4 ·
Replies
4
Views
2K
Linearly independent functions with identically zero Wronskian
  • · Replies 1 ·
Replies
1
Views
2K
If F is algebraically closed, show Alg ext is closed too
  • · Replies 1 ·
Replies
1
Views
1K