[Zhan lixing matrix theory exercise reference] Contents

Note:

1. Some of them are self-developed, while some are post-referenced.

2. If you have any good ideas and answers, please kindly inform me or reply to them on the corresponding Exercise answers page. I hope you can point out any errors.

3. After all, I have studied advanced algebra in college and want to learn more about matrix theory (matrix = magic). I will select this book first.

 

 

Chapter 1 prerequisites

[Zhan Xiang matrix theory exercise reference] exercise 1.1

1. set $ A_1, \ cdots, a_n $ as a positive real number, proof matrix $ \ Bex \ sex {\ frac {1} {a_ I + a_j }}_{ n \ times n} \ EEx $ semi-definite.

 

[Zhan Xiang matrix theory exercise reference] exercise 1.2

2. (oldenburgere) set $ A \ In M_n $, $ \ rock (a) $ to indicate the spectral radius of $ A $, that is, the creator of the modulus of the feature value of $ A $. proof: $ \ Bex \ vlm {k} a ^ K = 0 \ LRA \ rock (a) <1. \ EEx $

 

[Zhan Xiang matrix theory exercise reference] exercise 1.3

3. Prove that the numerical radius $ W (\ cdot) $ is a norm on $ M_n $.

 

[Zhan Xiang matrix theory exercise reference] exercise 1.4

4. it is proved that the relationship between the radius of the value $ W (\ cdot) $ and the spectral norm $ \ Sen {\ cdot} _ \ infty $ is as follows: $ \ Bex \ frac {1} {2} \ Sen {A }_{\ infty} \ Leq W (a) \ Leq \ Sen {A }_\ infty, \ quad A \ In M_n. \ EEx $

 

[Zhan Xiang matrix theory exercise reference] exercise 1.5

5. (Gelfand) set $ A \ In M_n $. Proof: $ \ Bex \ rock () = \ vlm {k} \ Sen {A ^ k} _ \ infty ^ \ frac {1} {k }. \ EEx $

 

[Zhan Xiang matrix theory exercise reference] exercise 1.6

6. set $ A \ in M _ {M, N }$, $ B \ in M _ {n, m} $. proof: $ \ Bex \ sex {\ BA {CC} AB & 0 \ B & 0 \ EA} \ mbox {And} \ sex {\ BA {CC} 0 & 0 \ \ B & Ba \ EA} \ EEx $ similar, this gives another proof of Theorem 1.14.

 

[Zhan Xiang matrix theory exercise reference] exercise 1.7

7. set $ a_j \ In M_n $, $ j = 1, \ cdots, M $, $ m> N $, and $ \ DPS {\ sum _ {j = 1} ^ m a_j} $ non-singular (reversible ). proof: $ s \ subset \ sed {1, 2, \ cdots, m} $ meet $ | S | \ Leq N $ and $ \ DPS {\ sum _ {J \ in S} a_j} $ is not singular.

 

[Zhan Xiang matrix theory exercise reference] exercise 1.8

8. It is proved that any compound matrix is similar to a matrix with all the equal corner elements.

 

[Zhan Xiang matrix theory exercise reference] exercise 1.9

9. prove to any compound arrays $ A $, $ \ Bex \ rock (a) \ Leq W (a) \ Leq \ Sen {A} _ \ infty. \ EEx $

 

[Zhan Xiang matrix theory exercise reference] exercise 1.10

10. matrix $ A = (A _ {IJ}) \ In M_n $ is called strictly Diagonal Dominant, if $ \ Bex | A _ {II} |> \ sum _ {J \ NEQ I} | A _ {IJ} |, \ quad I = 1, \ cdots, n. \ EEx $ proof: the strictly Diagonal Dominant Matrix is reversible.

 

[Zhan Xiang matrix theory exercise reference] exercise 1.11

11. (gersgorin disc theorem) $ \ sigma (a) $ is used to represent the set of feature values of $ A = (A _ {IJ}) \ In M_n $, note $ \ Bex d_ I =\sed {z \ In \ BBC; \ | Z-A _ {II} | \ Leq \ sum _ {J \ NEQ I} | A _ {IJ} |}, \ quad I = 1, \ cdots, n. \ EEx $ proof: $ \ Bex \ sigma (a) \ subset \ cup _ {I = 1} ^ n d_ I, \ EEx $ and if these disks $ d_ I $ have $ K and the remaining $ n-k $ do not intersect, then the union of the $ K $ disc exactly contains $ K $ feature values of $ A $.

 

[Zhan Xiang matrix theory exercise reference] exercise 1.12

12. (Sherman-Morrison-Woodbury formula) set $ A \ In M_n $, $ B, c \ in M _ {n, k} $ makes $ I + C ^ * a ^ {-1} B $ reversible, where $ I $ is a unit array. proof $ A + bc ^ * $ reversible and $ \ BEX (a + bc ^ *) ^ {-1} = a ^ {-1}-a ^ {-1} B (I + C ^ * a ^ {-1} B) ^ {-1} C ^ * a ^ {-1 }. \ EEx $

 

[Zhan Xiang matrix theory exercise reference] exercise 1.13

13. (Li-Poon) proof: Each real matrix can be written as a linear combination of $4 $ real positive matrix, that is, if $ A $ is a real matrix, the real positive matrix $ q_ I $ and the real number $ r_ I $, $ I = 1, 2, 4 $, make $ \ Bex a = r_1q_1 + r_2q_2 + r_3q_3 + r_4q_4. \ EEx $

 

[Zhan Xiang matrix theory exercise reference] exercise 1.14

14. if ing $ F: M_n \ To M_n $ sorts the elements of each matrix in $ M_n $ in a fixed mode, $ F $ is called a replacement operator. what kind of replacement operator does not change the feature value of the Matrix? Keep the rank unchanged?

 

 

 

[Zhan lixing matrix theory exercise reference] Contents