Eigenvalues of Invertible Matrix

Click For Summary
An invertible nxn matrix does not necessarily have n distinct eigenvalues; it can have fewer distinct eigenvalues. The key point is that an invertible matrix must not have 0 as an eigenvalue, which implies that all eigenvalues must be non-zero. The discussion highlights the distinction between the total number of eigenvalues and the number of distinct eigenvalues. The unit nxn matrix is indeed invertible, and its eigenvalues are all equal to 1. Understanding these concepts is crucial for correctly addressing the properties of eigenvalues in relation to matrix invertibility.
sonic25
Messages
8
Reaction score
0

Homework Statement


If A is an invertible nxn matrix, then A has n distinct eigenvalues. (TRUE/ FALSE)



Homework Equations





The Attempt at a Solution


True? We weren't really taught the concept of eigenvalues too well, but from what I can gather square matrices appear to have the same number of eigenvalues as their number of rows/columns. I'm not sure why though.
 
Physics news on Phys.org
Is the unit nxn matrix invertible? What are its eigenvalues?
 
  • Like
Likes 1 person
"number of eigenvalues" and "number of distinct eigenvalues" are two entirely different things. As far as being "invertible" is concerned the only thing you can say is that a matrix is invertible if and only if it does not have 0 as an eigenvalue.
 

Similar threads

Undergrad In 4x4 matrix when does row swapping not affect eigenvaluess?
  • · Replies 6 ·
Replies
6
Views
4K
How Do Eigenvalues and Eigenvectors Change for Matrix B = exp(3A) + 5I?
  • · Replies 14 ·
Replies
14
Views
2K
Prove the identity matrix is unique
  • · Replies 69 ·
3
Replies
69
Views
9K
Can You Solve for the Positive Definite Matrix with Eigenvalues 1 and 2?
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Getting eigenvalues of an arbitrary matrix with programming
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Generalized Eigenvalues of Pauli Matrices
  • · Replies 1 ·
Replies
1
Views
1K
Engineering How do I find the transition matrix of this dynamic system?
  • · Replies 18 ·
Replies
18
Views
3K
Prove true or false eigenvalue question
  • · Replies 3 ·
Replies
3
Views
2K
Linear Algebra: 2x2 matrix raised to the power of n
  • · Replies 9 ·
Replies
9
Views
7K
Linear algebra:Prove that the only invertible nxn idempotent matrix is I
  • · Replies 4 ·
Replies
4
Views
23K