I'm a first year PhD student and I was asked to think about a problem last week and I tried to work out some examples, did some computations, but I wasn't successful. Basically, I didn't really make any significant progress in the last week. Today I had a meeting with my supervisors and he immediately computed that example and it turns out that the computation was easy. There was no trick or anything, just plain and simple linear algebra. Now I'm filled with regret that I should have been able to do it on my own. Now I'm having this feeling of self-doubt. I feel like my supervisors might think less of me. What would you suggest I should do?
"You wouldn't worry so much about what others think of you if you realized how seldom they do" --- Eleanor Roosevelt.
You can relax; for experienced academics, the baseline expectation is that most new graduate students (including ourselves at that age) are/were basically incompetent. That is the reason we give you 4-5 years of training before we let you out in the world to do research on your own. It is unlikely you fall significantly below this low baseline expectation. I suspect you are just overestimating what academics think of the competence of the average grad-student, so don't worry; you are probably not significantly more incompetent than the others. What you observe as an "easy" computation done by your supervisor is the result of decades of training and experience in the field, which allows him to immediately identify classes of problems and solution methods that you do not yet grasp automatically.
In any case, I suggest you don't worry yourself about embarrassment with your supervisors and just work on the practical aspect of plugging skill gaps. Review the problem you had trouble with and identify why you were unable to identify the solution method. Do some practice problems if needed, and brush up generally on the material until you feel that you are able to comfortably solve problems of that general class. Your supervisors will/should tell you if you are significantly behind where you need to be at your stage of the PhD program, so if they haven't said anything, you are probably about where they expect you to be. If you are unsure, just ask your supervisor how you are tracking.
There is no need for any lingering sense of embarrassment. Most likely, your supervisor would have just been briefly reminded of the general incompetence of grad-students (and may have wistfully reflected on his own incompetence at your age), and then he would have gone onto something else and forgotten all about you.
When I was a first year Phd student in mathematics, I took a topology course. One theorem we covered was the Tietze extension theorem. I remember thinking about the proof for hours and hours, and feeling that it was completely opaque to me. I could verify each line, but I had no idea how anyone would have thought to put these ideas together in this particular way to prove the theorem. I couldn't make an outline of the proof. I would have had to memorize it basically verbatim to have a chance at proving it. I remember making a decision that this was too hard, that I would memorize the statement of the theorem and move on with my life.
Five years later I am close to getting my degree in a field which isn't really connected with point set topology. I have not thought about this theorem in years. A professor friend of mine is teaching this same topology class, we are having a conversation while he is walking to class, and I end up there. I decide to sit in. What do you know? The topic of the day is my old nemesis: the proof of the Tietze Extension Theorem.
My friend is asking the first year grad students questions very patiently, trying to get them to guess at how to prove the theorem. What I discover is that, at this point, the structure of the proof is obvious. Given the definitions of the structures involved, it feels like there is only one obvious strategy and verifying the details should be completely routine. It is!
Although I had not thought about point set topology for a few years, I had obviously grown a lot as a mathematician. What had been an impassable mountain had become a molehill.
I think this kind of experience is fairly common. You are literally learning to see a new world with new eyes. As a new grad student, you are like someone who has opened their eyes for the first time. The shapes and colors are overwhelming. Nothing makes sense. If you keep your eyes open, and keep playfully exploring, you will find that you orient to this new world and what is baffling now will become obvious.
Well, reminds me of some math exam I had in university including LA and my father asked me how it went. It was ok overall but I wasted considerable time on one task and got about half of it proved. So he asks me what it was, and I reiterate "Prove that the eigenvalues of antihermitian matrices are purely imaginary". So he does that thing where his view glances off to that secret blackboard in the sky and comes back and says "but that's trivial". What?!? So he writes down 5 lines and after secondguessing his notation (theoretical physicists use different notation than electrical engineers) and brooding over it for half an hour, I have to admit it's trivial.
As a side effect, you can probably wake me up in the middle of the night and ask me to prove that all eigenvalues of a hermitian/antihermitian/unitarian matrix are real/imaginary/something and I'll be able to do it before even waking up.
Stuff like that happens. If you had learnt it the regular way (chances are you did), you'd likely have forgotten about it again.
No way are you going to forget this again I'll bet. And it appears to be part of your trade's advanced tools.
So, good for you. At least if you didn't blow any deadline, and even then it would just be a temporary setback.
To build on Ben's answer, I'd just like to emphasize the "ask your supervisor" part.
Especially as an early grad student, it is important to develop a relationship with your advisor where you can admit things you don't understand, and ask them "dumb questions" so they will be better able to teach you and help fill those gaps. You are their student after all.
Graduate school and academia often have a highly deleterious effect on peoples' confidence, motivation, and feelings of self-worth. I have found personally that one effective strategy (among many!) to deal with this is to tell yourself that you are fully capable (e.g. I have no doubt you are fully capable of doing basic linear algebra), you just need to learn (1) what tools/information/skills to learn and (2) when and where to apply them. Those things are very often not learned by developing them from scratch by yourself, instead you will get them from textbooks, papers, and most importantly: by asking questions.
If you don't understand something, it's almost always best to just ask, rather than stay quiet and continue to not understand or even misunderstand. Your advisor should (in principle) be your first stop when it comes to asking questions. You should feel comfortable asking them about anything you don't know, because either (1) they know it and can explain it to you (or give you a reference to learn it), or (2) they don't know it and can point you in the direction of other resources where you could figure it out (which might mean new research opportunities!).
Having a good relationship with your advisor is important. Not every advisor is equal in this regard, but in the ideal situation you should be able to admit ignorance to them, ask them questions about anything you don't know, and maintain a clear channel of communication about their expectations and your progress as a grad student. (They can't accurately assess your progress if you hide things you don't know from them)
And of course asking questions doesn't only mean to your supervisor, I try to take any chance I get to ask questions about things I don't understand if I think the person I'm talking to might be able to clarify something for me (sometimes that means using Stack Exchange). Academic culture unfortunately doesn't do a great job of fostering positive learning environments where people are comfortable exposing their own ignorance in order to grow. But I assume that you have embarked on a graduate degree with a certain level of passion and drive to learn something, and I suggest taking any and every opportunity to do so, even if it means admitting you don't know something, even if you think that everyone else thinks it's trivial. You will often find that there is either a better perspective that makes it more "obvious" to you why it is trivial, or that it isn't so trivial as you had assumed. (Seriously, think of a classroom or conference setting, at least half the people probably don't know or were even wondering the same thing as you and were afraid to ask, so you're doing everyone a favor by asking)
Also, making friends and finding other grad student (or better yet, postdocs) who you can talk to and ask shameless questions to is hugely beneficial as well.
I won't say "don't worry about it" and I have anecdotes also, but let me give you a bit of perspective. Two main points.
The first is that "insight" in mathematics isn't general, but specific to some area(s). You can have great insight into one area and very little in another. For me, I had deep insight into classical topology and analysis, but very little in algebra. Ring Theory was especially difficult for me. I could follow proofs just fine and solve exercises, but couldn't go beyond that.
The second main point is related. It is a very different level of skill in math to be able to follow a proof than to generate one. An even higher level of insight is needed to be able to posit what might be true and is worth exploring.
It is the purpose of grad school (especially) to help you develop in both of these areas. You are still early in your studies, so it isn't especially worrisome at this point, though your advisor might have a different idea.
A minor point, but perhaps also important here, is that one doesn't normally simply "see" all possibilities for solving problems and sometimes early attempts actually prejudice against seeing others. Stuck in a rut, so to speak.
If this were your third year, I'd be more concerned. And if you couldn't follow the argument of the professor in solving the problem then I'd be concerned. But, the more you work at it, the easier you will likely find such things unless you work on harder and harder problems. In that case, there is a constant struggle to obtain insight - just as there is in obtaining wisdom. Carry on. The quest is worthwhile.
You didn't tell us anything about your supervisor. Does s/he already have a Ph.D.? If so, it would be silly to compare yourself to them, since they are by definition the one training you because they are the one who already has an advanced degree. Even a more senior graduate student has many hours of focus on research under their belt that you don't. Sometimes it takes lots of training to see the simple answers.
(There's a cartoon I can't find right now, I think xkcd, that shows how various people see some complicated theory. The undergrad sees a simple circle with a couple squiggles. The grad student sees a mass of equations. The postdoc sees a heap of experimental equipment. And the professor sees a simple circle with a couple squiggles.)
If you are worried about whether or not you have the capabilities to complete the program, compare yourself to other students at a similar stage in the program. Even then, give yourself some leeway - there will always be a lot of variation among students. Keep an eye on where you are relative to your cohort a few times a year.