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Label: style io color ar sp for on bs line3. (Bhatia-Davis) if $ A, B \ in M_n $ is A matrix, then $ \ bex \ rd (\ sigma (A), \ sigma (B )) \ leq \ sen {A-B} _ \ infty. \ eex $Proof: [see R. Bhatia, C. Davis, A bound for the spectral variation of a unitary operator, Linear and Multilinear Algebra, 15 (1984),-76.][Zhan Xiang matrix theory exercise reference] exercise 5.3

8. set $ k \ Leq m \ Leq N $. what kind of matrix $ A \ in M _ {m, n} $ each diagonal line exactly contains $ K $ zero elements? Answer: Theorem 2.5 (k \ "onig ), each diagonal line of $ A $ contains $ K $ zero elements $ \ LRA $ A $ has a zero submatrix of $ r \ times S $, $ R + S = N + k $; $ A $ a diagonal line contains $ k + 1 $ zero elements $ \ LRA $ any $ r \ times S $ level submatrix of $ A $ non-zero, $ R + S = N + k + 1 $. therefore, each diagonal line of $ A $ contains $ K $ zero

Label: Style Color SP on BS size nbsp C element3. Set $ a, B \ In M_n $, $ A $ positive definite, $ B $ semi-positive, and the logarithm element is positive, then $ A \ circ B $ positive.Proof: By Schur theorem, $ A \ circ B $ is semi-definite, and its feature value $ \ geq 0 $. to prove that $ A \ circ B $ is positive, only $ \ det (A \ circ B)> 0 $ ($ \ rA $ any feature value $> 0 $) must be proved ). this can be directly caused by the Oppenheim inequality $ \ Bex \ det (A \ circ B) \ geq \ de

Label: Style Color SP on BS amp AD Size DP 9. Note $ \ DPS {M = \ sex {n \ atop K }}$. Composite Matrix ing $ c_k (\ cdot): M_n \ To m_m $ is it a single shot? Is it a full shot? A: When $ k = 1 $, $ c_k (a) $ is each element of $ A $. therefore, $ c_k $ is a single shot or full shot. when $ k \ geq 2 $, generally, $ c_k $ is not a single shot, for example, $ \ Bex \ sex {\ BA {CCCC} 1 0 \ cdots 0 \ 0 \ cdots 0 \ vdots \ ddots \ vdots \ 0 0 \ cdots 0 \ EA }, \ quad \ sex {\ BA {CC

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? Answer: The replacement operator $ F $ keeps the feature value of the matrix unchanged. If only the replacement matrix $ p $ exists, $ \ Bex F (a) = pap ^ t, \ quad \ forall \ A \ In M_n; \ EEx $ or $ \ Bex F (a) = PA ^ TP ^ t, \ quad \ forall \ A \ In M

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) Proof: $ \ rA $: Based on Jordan standard theory, there is a reversible array $ p $, make $ \ Bex P ^ {-1} AP =\sex {\ BA {CCC} \ lm_1 * \\\\ ddots \\\ \ lm_n \ EA }, \ EEx $ and $ \ Bex P ^ {-1} a ^ Kp = \ sex {\ BA {CCC} \ lm_1 ^ K * \ ddots \\ \ lm_n ^ k \ EA }. \ EEx $ thus,

10. set $ A and B $ to the same-order semi-Definite Matrix, $0 \ Leq s \ Leq 1 $. proof: $ \ Bex \ Sen {A ^ Sb ^ s} _ \ infty \ Leq \ Sen {AB} _ \ infty ^ s. \ EEx $ Proof: (1 ). first, it is proved that the spectral norm of $ A $ is the maximum singular value of $ A $. in fact, $ \ beex \ Bea \ Sen {A} _ \ infty ^ 2 =\ Max _ {\ Sen {x} _ 2 = 1} \ Sen {ax} _ 2 ^ 2 \ =\ Max _ {\ Sen {x} _ 2 = 1} x ^ * a ^ * ax \ =\ Max _ {\ Sen {x }_ 2 = 1} x ^ * VV ^ * a ^ * u ^ * uavv ^ * x \ =\ Max

Label: Style Color SP on BS amp ad EF size 8. it is proved that each semi-positive definite matrix has a unique square root of the semi-positive definite number. That is, if $ A \ geq 0 $, there is a unique $ B \ geq 0 $ that satisfies $ B ^ 2 = A $. Proof: $ A \ geq 0 $ indicates the existence of U $, so that $ \ Bex U ^ * Au = \ diag (\ lm_1, \ cdots, \ lm_n ), \ quad \ lm_ I \ geq 0. \ EEx $ \ Bex B = U \ diag (\ SQRT {\ lm_1}, \ cdots, \ SQRT {\ lm_n}) U ^ *, \ EEx $ B \ geq 0 $, an

Label: Style Color SP on BS ad size as nbsp 5. Without the Weierstrass theorem, we can directly prove that the function operation (3.6) of the Hermite matrix has nothing to do with the specific spectral decomposition. Proof: $ h $ also has a spectral decomposition $ \ Bex H = V \ diag (\ lm_1, \ cdots, \ lm_n) V ^ *, \ EEx $ \ Bex w \ diag (\ lm_1, \ cdots, \ lm_n) = \ diag (\ lm_1, \ cdots, \ lm_n) W, \ quad W = V ^ * U. \ EEx $ therefore, $ \ Bex W = \ diag (W_1, \ cdots, w_s ), \ EEx $

Label: Style Color SP on bs ad Size nbsp C 1. Set $ A \ In M_n $. Prove that if $ AA ^ * = a ^ 2 $, $ A ^ * = A $. Proof: By Schur ry triangle theorem, $ U $ exists, making $ \ Bex a = u ^ * Bu, \ EEx $ where $ B = (B _ {IJ}) $ is the upper triangle array. so $ \ Bex U ^ * BB ^ * u = AA ^ * = a ^ 2 = u ^ * B ^ 2u \ rA BB ^ * = B ^ 2. \ EEx $ compare the diagonal yuan between two ends $ \ Bex | B _ {II} | ^ 2 + \ cdots + | B _ {In} | ^ 2 = B _{ II} ^ 2, \ Quad 1 \ Leq I \ Leq n. \ EEx $ whi

":Return ' death 'elif action = = "Slowly Place the bomb":Return ' Escape_pod 'ElseReturn ' The_bridge 'def escape_pod (self):Good_pod = Randint (1,5)Guess = Raw_input ("[Pod #]>>>")if int (guess)! = good_pod:Return ' death 'ElseExit (0)Class Engine (MAP):def __init__ (Self,start):Self.start = Startdef play (self):Next = Self.startWhile True:print "\ n-----------"GetAttr (Self,next)Next = Guest ()A_game = Engine ("Central_corridor")A_game.play () Remark: Beginner young novice, Welcome to the gre

Exercise: 1. display the lines starting with "S" of the/proc/meminfo file with no size difference; Grep-I s/proc/meninfo 2. display the rows ending with nologin in/etc/passwd; Grep nologin $/etc/passwd 3. the/etc/inittab starts with #, followed by one or more blank characters, and followed by any character rows; Grep -- color ^ #. */etc/inittab 4. the/etc/inittab displays the rows of a number (that is, a number between two colons; Grep: [0-9]:/etc/ini

{If (I! = K I! = J J! = K)/* Ensure that I, j, and K are different */Printf ("% d, % d, % d \ n", I, j, k );}} Aside from many other issues in this Code, the question "How many three numbers can be composed of different and no repeated numbers" is obviously ignored by programmers. Such code has no value. Therefore, we strongly recommend that you copy the question before writing the code.Some people may think that the question cannot be compiled. It doesn't matter. This can be an excellent com

Ansible (10) first exercise and ansible first exerciseSummary We have learned apt, yum, zypper, copy, and other modules. You must be bored.The following is a practical example.Actual case: Batch tomcat configuration Only the knowledge of the above modules is enough for us to orchestrate a Playbook to configure a large number of physical/virtual servers.Actual scenarios A company has applied for 100 cloud computing virtual machines and needs

, to Vertex y, up to the n-1 edge (to consider connectivity, each vertex up to 1 times), so the shortest possible X-to-y is up to n-1 relaxation operation (that is, the update length) should appear, if the first N times relaxation can also be optimal, then this diagram is definitely the existence of the ring (directly with Dijkstra can not be optimal, the existence of the ring will affect the existence of the optimal solution). Here gives the vertex when the English name of the currency, in ord

--subquery Exercises/*1, sub-query (internal query) before the main query (outer query) one execution completes 2, the result of the subquery is used by the main query 3, in the query list using subqueries, can only be a single-row single-column. See Exercise 2 4, do not use the ORDER BY statement in a subquery unless you are doing top N analysis, the efficiency of the factor query is low, and the ordering consumes resources. See

One, the treatment of stuttering.will be the string "I i i i i i i i I ...." I....... If you want to ....... ....... If you want to ... Learning to learn the breeze .... Breezy programming Cheng ..... Cheng "into" I want to learn programming "Analysis: First put .... Remove: Use "\\.+" mode, then replace the overlapping words with one: use "(.) \\1+ "mode.Code:1 PackageP02. Exercise;2 3 Public classDemo01 {4 Public Static voidMain (String args[

Linux OPS Practice-November 8, 2015-November 17 course Assignments (exercise) SummaryI. Homework (exercise) content:First, Iptables class workPremise: Input and output default policy is drop;1. The Web server that restricts the local host is not allowed to access in Monday; the rate of new requests cannot exceed 100 per second; The Web server contains the admin string and the page is not allowed; The Web se

Practice One:Pass three parameters to the script, the first is an integer, the second is an arithmetic operator, the third is an integer, the result of the calculation is displayed, and the two-bit precision is required to be preserved. Shaped like:./calc.sh 5/2#!/bin/bash#program:#练习2个整数进行基本算术运算#history 2016-10-21-14:10Path=/bin:/sbin:/usr/bin:/usr/sbin:/usr/local/bin:/usr/local/sbin:~/bashExport PATHResult= ' echo ' scale=2;$1 "| BC ' echo can continuously input parameters to the command behin

Lumbar intervertebral disc herniation is a common kind of orthopedic disease, usually lumbago, low back pain, Lumbar disc herniation may be caused by the disaster. So if the unfortunate suffering from lumbar intervertebral disc herniation, how should exercise? The following will introduce the self-exercise method of lumbar intervertebral disc herniationThe exercise