Machine Learning Foundations Quiz 2: 23 November 2025 (September 2025 term)

Consider the quadratic form
. If

, where
and
is a
matrix, which of the following is ?

Consider a matrix
of shape
. The SVD of
is given by
. The columns of are the

left singular vectors and the columns of
are the right singular vectors. The following details are given:

• The first left singular vector is
• The first right singular vector is

• The first singular value is

Find
.

Suppose
. What is the value of
when
is minimum?

For each sub-question, answer whether it is true or false.

The singular value decomposition exists only for square matrices.

For each sub-question, answer whether it is true or false.

If
is Hermitian, then eigenvectors corresponding to distinct eigenvalues are orthogonal.

Consider a unitary matrix
as given below:

Also,
and
.
Based on the above data, answer the given subquestions.

Enter the value of
.

Consider a unitary matrix
as given below:

Also,
and
.
Based on the above data, answer the given subquestions.

Enter the value of
.

Consider a unitary matrix
as given below:

Also,
and
.
Based on the above data, answer the given subquestions.

Consider
.
Based on the above data, answer the given subquestions.

Which of the following are the singular values of
?

Consider
.
Based on the above data, answer the given subquestions.

If
is a symmetric matrix, consider statements P and Q.

•
P: If
is an eigenvalue of
, then
is a singular value of
.

•
Q: If
is a singular value of
, then
is an eigenvalue of
.
Select the most appropriate option.

Hint: In the first sub-question,
is symmetric, and you should first look at its eigenvalues, which will lead you to some observations. Are these observations true for any symmetric matrix?

Consider the task of minimizing the function
. Let
be some arbitrary point in the plane other than the origin.
Based on the above data, answer the given subquestions.

Starting at
, run two steps of gradient descent with learning rate
to obtain

. If
, enter the value of
correct to two places after the decimal.

Note that the sequence of iterates will be:
.

Consider the task of minimizing the function
. Let
be some arbitrary point in the plane other than the origin.
Based on the above data, answer the given subquestions.

For what value of
will gradient descent reach the minimum after exactly one step? Enter your answer correct to one decimal place.

Consider the task of minimizing the function
. Let
be some arbitrary point in the plane other than the origin.
Based on the above data, answer the given subquestions.

Gradient descent is said to converge to the minimum $p^* = (x^*, y^*)$, if the point $p_t = (x_t, y_t)$ gets increasingly closer to $p^*$ as $t$ increases, that is, $\|p^* - p_t\|$ approaches zero as $t$ approaches infinity. Select all values of $\eta$ for which gradient descent converges to the minimum in this problem.

Based on the above data, answer the given subquestions.

If the covariance matrix of the dataset is
, find
.

Based on the above data, answer the given subquestions.

If the unit-norm eigenvector corresponding to the largest eigenvalue of
is
, find
.

Based on the above data, answer the given subquestions.

The data-points
and
are added to
to get
. PCA is run on . The

covariance matrix of
is
and the unit-norm eigenvector corresponding to the largest

eigenvalue of
is
. Which of the following is true?