I have a question regarding question 29 of the exam. In the solutions it is stated that the answer is TRUE however non-increasing order does not exclude a random ordering of the eigenvalues, or at least it does not to my understanding. I personally found the formulation a little misleading.
I am not sure i understand your question clearly.
a,b, c are sorted in non increasing order means that b<=a and c<=a. It is equivalent to say it is decreasing but neighboring quantities in the sequence can be equal.
I will try to explain myself a bit better. Assume the eigenvalues are in the set {1, 2, 3, 4}, then the order 4-3-2-1 is clearly non-increasing and in fact, it turns out to be decreasing as stated in the question. However, if the non-increasing requirement is analyzed at the full sequence level, then 4-3-1-2 is non-increasing (4>3>2) but at the same time, this sequence is also not decreasing (2>1).
In other words, in my opinion, the statement in the exam is TRUE only if the non-increasing requirement is imposed on EACH pair of sequential elements in the sequence, which was not stated.
In fact the sequence 4-3-1-2 is not non-increasing. The property is violated between 1 and 2.
Of course, the non-increasing property is imposed on each pair (by design, this is the definition of non-increasing for sequences).
I guess I misinterpreted the non-increasing concept with a not-increasing sequence in general. However, I do believe that an additional explanation of what was meant could have been useful to avoid confusion since that was not the main point of the question.
Exam 2021 Q29
Good morning,
I have a question regarding question 29 of the exam. In the solutions it is stated that the answer is TRUE however non-increasing order does not exclude a random ordering of the eigenvalues, or at least it does not to my understanding. I personally found the formulation a little misleading.
Thank you!
3
Hi,
I am not sure i understand your question clearly.
a,b, c are sorted in non increasing order means that b<=a and c<=a. It is equivalent to say it is decreasing but neighboring quantities in the sequence can be equal.
I will try to explain myself a bit better. Assume the eigenvalues are in the set {1, 2, 3, 4}, then the order 4-3-2-1 is clearly non-increasing and in fact, it turns out to be decreasing as stated in the question. However, if the non-increasing requirement is analyzed at the full sequence level, then 4-3-1-2 is non-increasing (4>3>2) but at the same time, this sequence is also not decreasing (2>1).
In other words, in my opinion, the statement in the exam is TRUE only if the non-increasing requirement is imposed on EACH pair of sequential elements in the sequence, which was not stated.
In fact the sequence 4-3-1-2 is not non-increasing. The property is violated between 1 and 2.
Of course, the non-increasing property is imposed on each pair (by design, this is the definition of non-increasing for sequences).
Thank you for the explanation,
I guess I misinterpreted the non-increasing concept with a not-increasing sequence in general. However, I do believe that an additional explanation of what was meant could have been useful to avoid confusion since that was not the main point of the question.
Thank you for your time and have a nice day.
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