Show that det(A) is the product of all the roots of the characteristic

k10sstar
Messages
1
Reaction score
0
Let A be an n x n matrix. Show that det(A) is the product of all the roots of the characteristic polynomial of A.
 
Physics news on Phys.org


State you definitions of det(A) and the characteristic polynomial of A, and show what you have done so far. If your definition of det(A) is the product of all the roots of the characteristic polynomial of A, you are done.
 


The quick proof is that any matrix is similar to either a diagonal matrix or a Jordan matrix with its eigenvalues on the main diagonal. In either case, the determinant is just the product of the numbers on the main diagonal- the product of the eigenvalues.

The difficulty with you not showing any attempt yourself is that we don't know what definitions and theorems you have available to prove this.
 

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 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Prove that the limit of this matrix expression is 0
Replies
5
Views
2K
I Question regarding fundamental region of a lattice
Replies
20
Views
2K
I About permutation acting on the Identity matrix
Replies
7
Views
3K
I Matrix rank in terms of determinant polynomial
Replies
14
Views
3K
MHB Relations between map and matrix
Replies
10
Views
1K
I Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
Replies
0
Views
2K
I Looking for a creative or quick method for finding roots in GF(p^n)
Replies
3
Views
1K
Having difficulty working out a Char. Pol. for Eigenvalues
Replies
1
Views
1K
I Finding a polynomial that has solution (root) as the sum of roots
Replies
13
Views
2K
MHB What Are the Possible Jordan Forms and Characteristic Polynomial of T?
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top