What is a good way to tackle difficult technical questions during graduate interview

I went into an engineering graduate school interview totally unprepared today and may have totally wrecked it.

The interview began like any other interview. The professor introduced his area of research, then I discussed my area of interest and academic background. I asked him a few questions regarding funding, number of graduate, thesis topic.

Then he said we should discuss some technical questions, which was not unreasonable, although I have never been asked to discuss technical questions with any other interviewers.

To begin he made me cite a number of theorems from linear algebra, I had some difficulty remembering the terminology but was in the ball park. Then he made me recite some theorems using formal proof "there exists...such that..." which really caught me off guard. He gave some other questions, but I had to ask him to clarify some of the terminologies he used in these questions which I don't think he was very eager to give out. Lastly, he made to write the closed form expression of some matrix operation which is a pain because I had to perform a bunch of mental calculations.

To be honest, I think I took too long to answer most of those questions and had trouble distinguish between the questions that should be addressed using intuition or mathematical derivation. The interviewer did not give obvious signals after I had answered a question correctly, this made me beat around the bush several times until he said my original answer was correct. Also, these questions were out of my current focus, despite having studied them at some point.

Can someone give me some tips on how to tackle questions in a graduate interval setting for which you think you will have difficult answering? What does the interviewer want to know in this case? How I can think step by step, how fast I can do this or if I was intuitive or made good use of analogy?

This may also be applicable to thesis deference, software interviews...