Master in Data Science (SMS, TU) Mathematics for Data Science Question Paper 2082 (Set pages 11-12; Second Assessment 2082) Nepal

For the matrix $A = \begin{pmatrix} -1 & 4 & 0 \\ 0 & 4 & 3 \\ 0 & 0 & 5 \end{pmatrix}$, find the eigenvalues and corresponding eigenvectors.

Orthogonally diagonalize the matrix $\begin{pmatrix} 3 & 4 \\ 4 & 9 \end{pmatrix}$.

Express the quadratic form $x^2 + 2xy + 4xz + 2y^2 + 10yz + 5z^2$ into $v^T A v$, where $v(x, y, z)$. Check your answer.

Find the singular values of the matrix $\begin{pmatrix} 7 & 1 \\ 5 & 5 \\ 0 & 0 \end{pmatrix}$.

Solve the following linear system by reducing the augmented matrix to the row echelon form:

a) Find the eigenvalues and eigenvectors of $A = \begin{pmatrix} 5 & -2 \\ 7 & -4 \end{pmatrix}$. State the effect of multiplying the eigenvector by the matrix A. b) Prove that if $v_1, v_2$ are the eigenvectors associated with the eigenvalues $\lambda_1, \lambda_2$ of a $2 \times 2$ symmetric matrix A respectively, then $A = \lambda_1 v_1 v_1^T + \lambda_2 v_2 v_2^T$.

Find a matrix $P$ that diagonalizes the matrix

Check your answer.

OR

Prove that a) If $A$ is an $n \times n$ matrix, $D = \text{diag}(d_1, \ldots, d_n)$, and $P$ is an invertible matrix such that $P^{-1}AP = D$, then for $1 \leq i \leq n$, the $i$-th column of $P$ is an eigenvector of $A$ corresponding to $d_i$. b) If $A$ is a real symmetric matrix, and $\lambda_1$ and $\lambda_2$ are distinct eigenvalues of $A$, then their corresponding eigenvectors $u$ and $v$ respectively are orthogonal.

Identify and sketch the conic whose equation is $5x^2 - 4xy + 8y^2 - 36 = 0$ by rotating the xy-axes to put the conic in standard position. Also, find the angle $\theta$ through which you rotated the xy-axes.

OR

Prove that a) If $A \in \mathbf{R}^{3\times 3}$ with a quadratic form in three variables, then there is a symmetric matrix $B$ in $\mathbf{R}^{3\times 3}$ such that $x^T A x = x^T B x$ for all $x \in \mathbf{R}^3$. b) If $A$ is an $n \times n$ symmetric matrix, then there is an orthogonal matrix $P$ such that the mapping defined by $x = Py$ transforms the quadratic form $x^T A x$ into a quadratic form $y^T D y$ with no cross-product term.

Determine the matrices U, D and V such that $A = U D V^T$ for the matrix

Solve $Ax = 0$, if $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$. Write the solution in vector form.