Which statement about eigenvalues and eigenvectors is not true?

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Discussion Overview

The discussion revolves around identifying which statement regarding eigenvalues and eigenvectors is not true among several provided options. The focus includes concepts of diagonalization, invertibility, and the existence of real eigenvalues in square matrices.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that statement b, which claims a matrix that can be diagonal is irreversible (or non-invertible), is the false statement.
  • Others clarify that "irreversible" may refer to "invertible," prompting questions about the definitions used.
  • One participant points out that the identity matrix is diagonal and invertible, suggesting that statement b must be false.
  • There is a discussion about whether a characteristic polynomial can have no real roots for a square matrix, with some participants indicating that they believe this is possible.
  • Examples are provided, including a specific 2x2 matrix, to illustrate points about diagonalization and invertibility.

Areas of Agreement / Disagreement

Participants express differing views on which statement is false, particularly regarding statement b. There is no consensus on the correctness of the statements, and multiple competing views remain.

Contextual Notes

Participants discuss the definitions of terms like "invertible" and "diagonalizable," which may lead to varying interpretations of the statements. The discussion includes unresolved questions about the implications of characteristic polynomials and the conditions under which matrices can be diagonalized.

Yankel
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One more question please...

which one of these statements is NOT true (only one can be false):

a. a square matrix nXn with n different eigenvalues can become diagonal.

b. A matrix that can be diagonal is irreversible.

c. Eigenvectors that correspond to different eigenvalues are linearly independent.

d. There are square matrices with no real eigenvalue.

I think that b is correct...

thanks
 
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Yankel said:
One more question please...

which one of these statements is NOT true (only one can be false):

a. a square matrix nXn with n different eigenvalues can become diagonal.

b. A matrix that can be diagonal is irreversible.

c. Eigenvectors that correspond to different eigenvalues are linearly independent.

d. There are square matrices with no real eigenvalue.

I think that b is correct...

thanks

What does "irreversible" mean in this context? Do you mean invertible?
 
yes, sorry, I mean not invertible, meaning, has no inverse.
 
consider this: the matrix I is diagonal, yet it is invertible, so...
 
I have eliminated most answers out, so I am left with 2...

the first is the invertible thing, and the second is that there are no squared matrices with no real eignvalue.

According to definition: D=P^-1 * A * P

So does A need to be invertible ? Why ?

Is it possible that a characteristic polynomial will have no real roots for a squared matrix ? I think so...
 
Yankel said:
I have eliminated most answers out, so I am left with 2...

the first is the invertible thing, and the second is that there are no squared matrices with no real eignvalue.

According to definition: D=P^-1 * A * P

So does A need to be invertible ? Why ?

Is it possible that a characteristic polynomial will have no real roots for a squared matrix ? I think so...

How about
$$\begin{bmatrix}0 &-1\\ 1 &0\end{bmatrix}?$$
 
Ackbach said:
How about
$$\begin{bmatrix}0 &-1\\ 1 &0\end{bmatrix}?$$
And just to be clear about the second option, you mean to write that

b. A matrix that can be diagonalized is non-invertible (or singular).

Is that the correct b. option?
 
I don't know which one is correct, but I am quite convinced now it's b, your example with the 2X2 matrix was good, I think it's done, thanks !
 
Yankel said:
I don't know which one is correct, but I am quite convinced now it's b, your example with the 2X2 matrix was good, I think it's done, thanks !

You're welcome!
 
  • #10
Yankel said:
I don't know which one is correct, but I am quite convinced now it's b, your example with the 2X2 matrix was good, I think it's done, thanks !

as i pointed out before, I is a diagonal matrix (thus it is "diagonalizable" you don't even have to do anything!) that is invertible (I is its own inverse), so b. must be false.
 

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