Timeline for Is it true that a 2x2 matrix is diagonalizable iff it has two distinct eigenvalues?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jul 6, 2020 at 1:49 | comment | added | Ned | @J.C.VegaO yes you are right -- that IFF statement is only correct if you eliminate the scalar matrices $cI_2$ from consideration. | |
| Jul 6, 2020 at 0:39 | comment | added | some_math_guy | @ned Actualy we had a matrix with unknown entries a11,a12,a21,a22 an we had to find the conditions for it not to be diagonalizable, he said the characteristic polynomial should have a double root, that is the matrix should have only a double eigenvalue, and wrote the equation ,but he said nothing about the matrix not being diagonal nor write it down. This adds an extra condition: a12 and a21 should not be both zero, don't you agree then that the statement was incomplete? | |
| Jul 6, 2020 at 0:26 | comment | added | Ned | @J.C.VegaO my guess was that the professor meant "if you are looking at a 2x2 matrix, either you can see that it's diagonal already, or if it's not, it can be diagonalized IFF it has two distinct eigenvalues." | |
| Jul 6, 2020 at 0:04 | comment | added | some_math_guy | @ned I didn't said non-diagonal, but non-diagonalizable, that is the professor said that in order to be non-diagonalizable, it should have two disctinct eigenvalues, and I asked if a matrix in general was diagonalizable iff it had 2 distict eigenvalues. From your statement I deduce that what the professor said was not entirely correct because he didn't said non-diagonal matrices, and my counterexampe shows we can have a matrix with a double eigenvalue that is diagonalizzable, if the matrix is already diagonal | |
| Jul 5, 2020 at 23:51 | comment | added | Ned | The OP is correct in saying that a 2x2 NON-DIAGONAL matrix is diagonalizable IFF it has two distinct eigenvalues, because a 2x2 diagonal matrix with a repeated eigenvalue is a scalar matrix and is not similar to any non-diagonal matrix. | |
| Jul 5, 2020 at 23:41 | comment | added | some_math_guy | So if I am correct the positive statement should be: If A is diagonalisable, then it has one eigenvalue of multiplicity 2, that is it is just an if not an iff. | |
| Jul 5, 2020 at 23:36 | comment | added | some_math_guy | What should the correct negation of my prof.'s statement be? | |
| Jul 5, 2020 at 23:31 | history | answered | John Hughes | CC BY-SA 4.0 |