Trace of a matrix equals sum of its determinants?

Click For Summary
The trace of a diagonalizable matrix equals the sum of its eigenvalues because the trace is defined as the sum of the diagonal elements, which correspond to the eigenvalues in a diagonal matrix. The discussion highlights the importance of understanding the algebraic properties of the trace, particularly its similarity invariance, meaning it remains constant under similarity transformations. This property allows for the conclusion that the trace of any diagonalizable matrix is equal to the sum of its eigenvalues. The participants express a need for a formal proof and clarification on the trace of diagonal matrices. Overall, the relationship between the trace and eigenvalues is a fundamental concept in linear algebra.
samspotting
Messages
86
Reaction score
0
If a matrix is diagonalizable, how does its trace equal the sum of its eigenvalues?

I can't find a proof for this anywhere.
 
Physics news on Phys.org
What is the trace of a diagonal matrix? What are the algebraic properties of trace?
 
Ah I see, the trace is similarity invariant. thanks
 

Thread 'Confusion about the Moyal-Weyl twist'

I am studying the mathematical formalism behind non-commutative geometry approach to quantum gravity. I was reading about Hopf algebras and their Drinfeld twist with a specific example of the Moyal-Weyl twist defined as F=exp(-iλ/2θ^(μν)∂_μ⊗∂_ν) where λ is a constant parametar and θ antisymmetric constant tensor. {∂_μ} is the basis of the tangent vector space over the underlying spacetime Now, from my understanding the enveloping algebra which appears in the definition of the Hopf algebra...
View full post »

Similar threads

Graduate Trace of the inverse of matrix products
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Matrix rank in terms of determinant polynomial
  • · Replies 14 ·
Replies
14
Views
3K
Graduate How to take a matrix outside the diagonal operator?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
2K
Undergrad Spectral theorem for Hermitian matrices-- special cases
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Similarity transformation changing the determinant to 1
  • · Replies 20 ·
Replies
20
Views
3K
What is the practical application of the trace of a square matrix?
  • · Replies 5 ·
Replies
5
Views
4K
Undergrad Unravelling Structure of a Symmetric Matrix
  • · Replies 5 ·
Replies
5
Views
3K
Multipole Expansion: Show that the quadrupole moment is symmetric and that the trace vanished
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad If L is diagonalizable, algebraic & geometric multiplicities are equal
  • · Replies 4 ·
Replies
4
Views
3K