Master in Data Science (SMS, TU) Mathematics for Data Science Question Paper 2081 Nepal

If $\lambda = 1, 5$ eigenvalues of the matrix $\begin{pmatrix} 7 & 4 \\ -3 & -1 \end{pmatrix}$, find a basis for the eigenspace corresponding to each eigenvalue.

Find the maximum value of $9x_1^2 + 4x_2^2 + 4x_3^2$ subject to the constraints $x^T x = 1$ and $x^T u_1 = 0$, where $u_1 = (1, 0, 0)$. Find $x$ where it is attained. Here, $u_1$ is a unit eigen vector corresponding to the greatest eigenvalue $\lambda = 9$ of the matrix of the quadratic form.

Find the singular values of the matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \\ -1 & 1 \end{pmatrix}$.

Consider the quadratic form $Q(x) = 2x_1^2 + 4x_1 x_2 - 4x_1 x_3 - x_2^2 + 8x_3 x_2 - x_3^2$. Decide whether this quadratic form is positive, negative or indefinite.

Determine if the following homogeneous system has a nontrivial solution. Then describe the solution set.

$3x_1 + 5x_2 - 4x_3 = 0, \quad -3x_1 - 2x_2 + 4x_3 = 0, \quad 6x_1 + x_2 - 8x_3 = 0.$

Consider the matrix: $A = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}$.

a) What can we say about the action of $A$ on an arbitrary vector?

b) What are examples of eigenvalues and eigenvectors of this matrix?

c) What does the discussion for this example illustrate?

OR

a) Let $v_1, v_2$ be the eigenvectors associated with the eigenvalues $\lambda_1, \lambda_2$ of a $2 \times 2$ symmetric matrix $A$ respectively. Prove that if $A = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix}$ and $V = (v_1 v_2)$, then $A = V A V^T$.

b) Find all $2 \times 2$ matrices $A$ which admit the normalized eigenvectors $v_1 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ and $v_2 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}$ with the corresponding eigenvalues $\lambda_1$ and $\lambda_2$.

a) Let $A$ be an $n \times n$ matrix. Prove that if $A$ has $n$ linearly independent eigenvectors, then $A$ is diagonalizable.

b) Show that the matrix $A = \begin{pmatrix} 2 & -1 \\ 1 & 4 \end{pmatrix}$ is not diagonalizable.

a) Prove that if $A$ is a symmetric $n \times n$ matrix and $B_A(v, w) = v^T A w$, then $B_A(v, w)$ is linear in the first variable $v$.

b) Write the quadratic form $10x_1^2 - 8x_1 x_2 + 4x_2^2$ as $x^T A x$. Transform it into a quadratic form without the cross product term using eigenvalues and eigenvectors.

Find an SVD of the matrix $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 2 \end{pmatrix}$.

OR

a) Prove that if $A$ is an $m \times n$ matrix, then all the eigenvalues of $A^T A$ are non-negative.

b) Find the eigenvalues and eigenvectors of $A^T A$ where $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \\ -2 & 1 \end{pmatrix}$.

What is reduced row echelon form? Illustrate with an example of an augmented matrix of order $4 \times 5$. Solve the following linear system by placing the augmented matrix in reduced row echelon form.

$2x + y - z = 2, \quad 4x + 3y + 2z = -3, \quad 6x - 5y + 3z = -14.$