MATEMATISKA INSTITUTIONEN Matristeori的数学专业矩阵论matrix theory科目的代考成功案例，学生在我们的帮助下取得了满意的成绩~

Hand in solutions of 7 from 8 problems below ( 6 problems if you applied for 6 course points, e.g. old LTH students). For a passing grade ( $3.5$ for $7.5$ points course and $3.0$ for 6 points course), at least one problem of the last four have to be solved. Credits can be given for partially solved problems. Write your solutions (in English or Swedish) neatly and explain your calculations. Both the content and the format of your solutions, and also how difficult problems you choose, will affect your grade.

The exam (in any readable format, preferably pdf) should be send by e-mail at the latest January 16. Write your name, section-year (or subject for Ph.D-students), idnumber, phone number and email address on the first page, and write your name on each of the following pages. Write also preferable time for the oral part. Write explicitly if you are student (LTH or NF) or graduate student and if you have applied to 6 points course. The result will be send to the email address written on the first page. The oral part of the exam should take place in January-February (depending on your schedule).

You may use any books and computer programs (e.g. Matlab and Maple), but it is not permitted to get help from other persons, including from internet. Programs and long calculations can be submitted by e-mail. You do not need to explain the elementary matrix operation such as matrix multiplications made by the computer, but should explain steps in more complicated calculations such as jordanization. You can check in such cases your results by computer but do not need to submit such calculations. Ask if you are not sure!

You can use any theorem in the book without proof but not exercises without explaining the solutions.
All matrices below are complex matrices if it is not specified explicitly.
Good luck!
Victor Ufnarovski
[email protected]

Problem 1.

1. Let
$$A=\frac{\pi}{6}\left(\begin{array}{rr} -2 & -1 \ 0 & -2 \end{array}\right) .$$
Find $\cos A, \sin A$ and check that $\sin (-A)=-\sin A$ and $\cos (-A)=\cos A$. Are these identities true for all square matrices $A$ ?

Problem 2.

Let $k \geq 1$. Show that a matrix has rank $r \leq k$ if and only if it can be written as a sum of $k$ matrices of rank 1 .

Problem 3.

For arbitrary integers $m, n>0$ find the singular values for the $m \times n$ matrix $A$, in which all elements are equal to 7 . For $m=4, n=3$ and $B=(-1,2,3,-4)^T$ also find the Moore-Penrose pseudoinverse $A^{+}$and a vector $X$ such that $|A X-B|_2$ is minimal. Is the vector $X$ unique?

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