What is the Connection Between Matrix Trace and Endomorphism?

  • Thread starter Thread starter tamintl
  • Start date Start date
  • Tags Tags
    Trace
tamintl
Messages
74
Reaction score
0
Consider the 4x4 matrices
A =
(1 2 3 4)
(5 6 7 8)
(9 10 11 12)
(13 14 15 16)B=
(1 2 3 4)
(8 5 6 7)
(11 12 9 10)
(14 15 16 13)

The question I was asked was the following: Show that there does not exist an endomorphism f: ℝ4 -> ℝ4 and basis 'a' and 'b' of R^4, such that A = a[f]a and B=b[f]b.

I have read in my notes and found that if the traces of the two matrices are not the same then they cannot represent the same endomorphism.

I am struggling to see the intuition behind this though.

Can anyone shed some light?

Many thanks
 
Physics news on Phys.org
The trace of a linear transformation is the sum of the eigenvalues of the matrix, and so is independent of the choice of basis.

Alternatively, you can use the fact that Tr(AB) = Tr(BA) to show that if you conjugate a matrix by another matrix the trace is unchanged.
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread '##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?'

I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
View full post »

Similar threads

I Matrix diagonalization algorithm
Replies
5
Views
2K
MHB Determine the cycle decomposition of the permutations
Replies
28
Views
3K
MHB Solve Bracelet Problem w/14 Beads (Red, White, Blue)
Replies
1
Views
2K
I Jacobi matrix diagonalization problems
Replies
5
Views
2K
I Solve the system of linear equations
Replies
2
Views
3K
I Building a coefficient matrix for a system of equations
Replies
7
Views
2K
MCNP Problem - Bad character in column 2
Replies
7
Views
4K
MHB Define matrix to get a row operation of type 1
Replies
2
Views
1K
MHB 311.2.2.6 use inverse matrix to solve system of equations
Replies
2
Views
1K
MHB *elements and generators of U(14)
Replies
2
Views
6K

Hot Threads

Recent Insights

Back
Top