🔬 Tutorial problems epsilon \epsilon

🔬 Tutorial problems epsilon \(\epsilon\)#

Note

This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left out are for additional practice on your own.

\(\epsilon\).1#

What is the rank of the \(N \times N\) identity matrix \(I\)?

What about the upper-triangular matrix which diagonal elements are 1?

Check all relevant definitions

\(\epsilon\).2#

Show that if \(T: \mathbb{R}^N \to \mathbb{R}^N\) is nonsingular, i.e. linear bijection, the inverse map \(T^{-1}\) is also linear.

Check all relevant definitions

\(\epsilon\).3#

Chose an orthonormal basis in \(\mathbb{R}^3\) which is not canonical basis \(\{e_1,e_2,e_3\}\) and verify by direct computation that the transformation matrix is orthogonal.

To come up with an orthonormal basis in \(\mathbb{R}^3\) think first of three orthogonal vectors (directions from the origin), then write them as vectors, and normalize each to length 1.

\(\epsilon\).4#

For each of the linear maps defined by the following matrices

\[\begin{split} T_1 = \begin{pmatrix} 4/3 & -2/3 & 0 \\ -1/3 & 5/3 & 0 \\ 0 & 0 & -1 \end{pmatrix} \end{split}\]
\[\begin{split} T_2 = \begin{pmatrix} 4 & 0 & 1 \\ -2 & 1 & 0 \\ -2 & 0 & 1 \end{pmatrix} \end{split}\]
\[\begin{split} T_3 = \begin{pmatrix} 5 & 0 & 1 \\ 1 & 1 & 0 \\ -7 & 1 & 0 \end{pmatrix} \end{split}\]
\[\begin{split} T_4 = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & -1 \\ 0 & -1 & 3 \end{pmatrix} \end{split}\]

perform the following tasks:

  1. Find eigenvalues

  2. Find at least one eigenvector for each eigenvalue

  3. Form a new basis from the eigenvectors (normalized or not)

  4. Compute the transformation matrix to the new basis

  5. Find the matrix \(T\) in the new basis and verify that it is diagonal

See example in the lecture notes

\(\epsilon\).5#

Compute 10th power of the following matrix

\[\begin{split} A = \begin{pmatrix} 2 & 1 & 3\\ 1 & 2 & 3\\ 1 & 1 & 2 \end{pmatrix} \end{split}\]

No way you should compute tenth power directly. Consider diagonalization of \(A\).

\(\epsilon\).6#

A stochastic matrix is a square matrix, whose rows sum up to 1.

Consider the following \(n \times n\) stochastic matrix:

\[\begin{split} A_n = \left( \begin{array}{cccccc} \alpha_1 & 0 & 0 & \dots & 0 & 1-\alpha_1 \\ 0 & \alpha_2 & 0 & \dots & 0 & 1-\alpha_2 \\ 0 & 0 & \alpha_3 & \dots & 0 & 1-\alpha_3 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & \alpha_{n-1} & 1-\alpha_{n-1} \\ 1-\alpha_n & 0 & 0 & \dots & 0 & \alpha_n \end{array} \right), \end{split}\]

where \(\alpha_i \in [0,1]\) for \(i=1,2,\dots,n\).

Show that the maximum eigenvalue of \(A_n\) is 1 for all \(n \in \mathbb{N}\).

Both direct proof and proof my mathematical induction will work. In both cases it is worth starting with the simple case of \(n=2\).

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