Research Stream Preference: Intelligent Systems (background subject: linear algebra and probability theory)

There were three professors in the panel, only two of them asked me questions. I will refer them by I1 and I2.

I1: Read out my application, mentioned my GATE score and rank, department preferences, labs selected, and asked about my BTech CGPA

He asked me why I did not apply for direct PhD and explained me benefits of doing direct PhD.

I1: What are the subjects that you have prepared for the interview?

Me: Sir, I have prepared background subjects linear algebra and probability, and familiar with subjects in GATE syllabus.

I1: I will ask you questions in probability first. Tell me if you have referred any standard book for probability?

[I mentioned the Sheldon Ross book]

I1: There are two types of random variables: discrete and continuous. Can you tell me what is the difference between the two random variables?

Me: For discrete random variables the values that the random variable (X) can take are countably finite and for continuous random variables the values are uncountable.

I1: (He was not impressed with my answer) Are you sure that is the correct statement? And is it written in Sheldon Ross book?

I1: Why do we name it continuous random variable? Why not call it uncountable random variable?

(We had discussion about this for about 5 minutes then he said that we call it 'continuous' because the cumulative distribution function for the random variable is a continuous graph.)

I1: Coming to normal random variable, can the peak of the bell curve have value greater than 1?

Me: The sum of probabilities in the range will give the total probability = 1. So total area under the curve should be 1. But I am not sure if peak could be greater than 1.

(Interviewer was not convinced with my answer. He gave me a few hints but I couldn't answer it well)

I1: What will be the value of normal random variable for $X = x$

Me: $P(X=x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{-(x-\mu)^2}{2\sigma^2}}$

I1: Right. What will be the maximum value for this? (assuming mean=0 and std=1)

Me: $\frac{1}{\sqrt{2\pi}}$ at x=0.

I1: Tell me about the binomial random variable.

Me: Told

I1: Can you say that binomial random variable is a limiting case of normal random variable? (If n is very large then will the binomial random variable become similar to normal random variable?) Prove it.

Me: I said yes but I couldn't prove the statement.

Now, the other professor took over and asked me questions on linear algebra

I2: What are eigenvalues and eigenvectors, can you define them?

Me: Sir, these are vectors whose direction remains unchanged after transformation. They might squeeze or stretch but the direction remains unchanged.

I2: Can you write the equation that you would use to calculate the eigenvalues and eigenvectors. And also explain the above definition with respect to that equation.

Me: Told

I2: Do these eigenvalues exist for every matrix mxn ?

Me: No sir, they are defined for only square matrices.

I2: Yeah, can you tell me why?

Me: In $Ax = \lambda x$ if A is rectangular matrix (m, n) and x is a (n, 1) column vector then Ax would be m-dimensional column vector (m, 1) which is different from RHS, i.e $\lambda x$ is a n-dimensional column vector (n, 1).

I2: Can you workout the above equation and show me how you will go about calculating the eigenvalues.

Me: $Ax = \lambda x$

$Ax = \lambda I x$

$(A-\lambda I)x=0$

$det(A-\lambda I)=0$

Now, we will solve for $\lambda$

I2: Okay, I understand the first two steps but why have you made $det(A-\lambda I)=0$ in the third step.

Me: Since we have $(A-\lambda I)x=0$. To solve this equation, x represents some combination of column vectors of $(A-\lambda I)$ that equate to zero. So the column vectors are not independent and hence the rank would be less than n. That would mean the determinant of (n x n) matrix should be zero if we want a non-trivial solution.

I2: Okay, now suppose I take transpose of A. Will the eigenvalues change?

Me: No sir. If $det(A-\lambda I)=0$ then $det((A-\lambda I)^T) = 0$ because the determinant remains same after transpose. Now, $det(A^T-\lambda I)=0$. Hence, the eigenvalues $\lambda$ will remain same.

I2: Okay, so you said that determinant remains unchanged after transpose. Is that true for every matrix?

Me: Yes, sir

I2: Do you know about row echelon form or reduced row echelon form. Can you name some common matrix operations?

Me: Yes sir, we can exchange two rows/columns, we can subtract/add multiple of some row/column and add to the other row/column.

I2: What would happen to the determinant if we apply these transformations.

Me: Sir, the determinant would remain unchanged.

I2: Can you take a simple matrix and show with a simple operation that determinant remains unchanged?

Me: I tried to work out a proof with a simple 2x2 matrix but couldn't clearly do it. He did give me some hint. Later I showed him an example that in a 2x2 matrix if we do $R_2=R_2-3R_1$ then the determinant value remains same.

(He was looking for a formal proof but I couldn't do it due to panic so I just showed him a simple worked out example by taking a 2x2 matrix)

The interview ended here, it lasted about 45 minutes.

My suggestion to future aspirants will be to prepare the topics from standard books right from the start of their GATE preparation. The interviewers focus on concepts and proofs which are very well explained in the standard books.

Verdict: Not Selected

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