# Đại số ma trận ứng dụng trong giải tích mạng

 Page 1 / 3
Trong chương này ta nhắc lại một số kiến thức về đại số ma trận thông thường được ứng dụng trong giải tích mạng.

## Kí hiệu ma trận:

Ma trận chữ nhật A kích thước m x n là 1 bảng gồm m hàng và n cột có dạng sau:

$\begin{array}{cc}A& \mid \begin{array}{cccc}{a}_{\text{11}}& {a}_{\text{12}}& \text{.}\text{.}\text{.}& {a}_{1n}\\ {a}_{\text{21}}& {a}_{\text{22}}& \text{.}\text{.}\text{.}& {a}_{2n}\\ \text{.}\text{.}\text{.}& \text{.}\text{.}\text{.}& \text{.}\text{.}\text{.}& \text{.}\text{.}\text{.}\\ {a}_{\mathrm{m1}}& {a}_{\mathrm{m2}}& \text{.}\text{.}\text{.}& {a}_{\text{mn}}\end{array}\mid \end{array}=\left[{a}_{ij}\right]$

Nếu m = 1 và n>1 thì A gọi là ma trận hàng hoặc vectơ hàng.

Ngược lại n = 1 và m>1 thì A gọi là ma trận cột hoặc vectơ cột.

Ví dụ: $A=\mid \begin{array}{c}2\\ 1\\ 3\end{array}\mid$ $A=\mid \begin{array}{ccc}2& 3& 1\end{array}\mid$

## Các dạng ma trận:

Ma trận vuông: Là ma trận có số hàng bằng số cột (m = n).

Ví dụ:

$\begin{array}{cc}A& \mid \begin{array}{ccc}{a}_{\text{11}}& {a}_{\text{12}}& {a}_{\text{13}}\\ {a}_{\text{21}}& {a}_{\text{22}}& {a}_{\text{23}}\\ {a}_{\text{31}}& {a}_{\text{32}}& {a}_{\text{33}}\end{array}\mid \end{array}$

Ma trận tam giác trên: Là ma trận vuông mà các phần tử dưới đường chéo chính aị j của ma trận bằng 0 với i>j.

$\begin{array}{cc}A& \mid \begin{array}{ccc}{a}_{\text{11}}& {a}_{\text{12}}& {a}_{\text{13}}\\ 0& {a}_{\text{22}}& {a}_{\text{23}}\\ 0& 0& {a}_{\text{33}}\end{array}\mid \end{array}$

Ma trận tam giác dưới: Là ma trận vuông mà các phần tử trên đường chéo chính aịj của ma trận bằng 0 với i<j.

$\begin{array}{cc}A& \mid \begin{array}{ccc}{a}_{\text{11}}& 0& 0\\ {a}_{\text{21}}& {a}_{\text{22}}& 0\\ {a}_{\text{31}}& {a}_{\text{32}}& {a}_{\text{33}}\end{array}\mid \end{array}$

Ma trận đường chéo: Là ma trận vuông nếu tất cả các phần tử trên đường chéo chính khác 0, còn các phần tử khác ngoài đường chéo chính của ma trận bằng 0 (aịj = 0 với $i\ne j$ ).

$\begin{array}{cc}A& \mid \begin{array}{ccc}{a}_{\text{11}}& 0& 0\\ 0& {a}_{\text{22}}& 0\\ 0& 0& {a}_{\text{33}}\end{array}\mid \end{array}$

Ma trận đơn vị: Là ma trận vuông mà tất cả các phần tử trên đường chéo chính của ma trận bằng 1 còn tất cả các phần tử khác bằng 0 (aij = 1 với i = j và aịj = 0 với $i\ne j$ ).

$\begin{array}{cc}U& \mid \begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\mid \end{array}$

Ma trận không: Là ma trận mà tất cả các phần tử của ma trận bằng 0.

Ma trận chuyển vị: Là ma trận mà các phần tử aịj = aji (đổi hàng thành cột và ngược lại).

$\begin{array}{cc}A& \mid \begin{array}{cc}{a}_{\text{11}}& {a}_{\text{12}}\\ {a}_{\text{21}}& {a}_{\text{22}}\\ {a}_{\text{31}}& {a}_{\text{32}}\end{array}\mid \end{array}$ $\begin{array}{cc}{A}^{T}& \mid \begin{array}{ccc}{a}_{\text{11}}& {a}_{\text{21}}& {a}_{\text{31}}\\ {a}_{\text{12}}& {a}_{\text{22}}& {a}_{\text{32}}\end{array}\mid \end{array}$

Cho ma trận A thì ma trận chuyển vị kí hiệu là At, AT hoặc A’

Ma trận đối xứng: Là ma trận vuông có các cặp phần tử đối xứng qua đường chéo chính bằng nhau aịj = aji.

Ví dụ:

$\begin{array}{cc}A& \mid \begin{array}{ccc}1& 5& 3\\ 5& 2& 6\\ 3& 6& 4\end{array}\mid \end{array}$

Chuyển vị ma trận đối xứng thì AT = A, nghĩa là ma trận không thay đổi.

Ma trận xiên - phản đối xứng: Là ma trận vuông có A = - AT. Các phần tử ngoài đường chéo chính tương ứng bằng giá trị đối của nó (aịj = - aji) và các phần tử trên đường chéo chính bằng 0.

Ví dụ:

$\begin{array}{cc}A& \mid \begin{array}{ccc}0& 5& -3\\ -5& 0& 6\\ 3& -6& 0\end{array}\mid \end{array}$

Ma trận trực giao: Là ma trận có ma trận chuyển vị chính là nghịch đảo của nó. (AT .A = U = A .AT với A là ma trận vuông và các phần tử là số thực).

Ma trận phức liên hợp: Là ma trận nếu thế phần tử a + jb bởi a - jb thì ma trận mới A* là ma trận phức liên hợp.

Cho ma trận A thì ma trận phức liên hợp là A*

$A=\mid \begin{array}{cc}\mathrm{j3}& 5\\ 4+\mathrm{j2}& 1+\mathrm{j1}\end{array}\mid$ $\begin{array}{cc}{A}^{}& \mid \begin{array}{cc}-\mathrm{j3}& 5\\ 4-\mathrm{j2}& 1-\mathrm{j1}\end{array}\mid \end{array}$

-Nếu tất cả các phần tử của A là thực, thì A = A*

-Nếu tất cả các phần tử của A là ảo, thì A = - A*.

Ma trận Hermitian (ma trận phức đối): Là ma trận vuông với các phần tử trên đường chéo chính là số thực còn các cặp phần tử đối xứng qua đường chéo chính là những số phức liên hợp, nghĩa là A = (A*)t.

$\begin{array}{cc}A& \mid \begin{array}{cc}4& 2-\mathrm{j3}\\ 2+\mathrm{j3}& 5\end{array}\mid \end{array}$

Ma trận xiên - Hermitian (ma trận xiên - phức đối): Là ma trận vuông với các phần tử trên đường chéo chính bằng 0 hoặc toàn ảo còn các cặp phần tử đối xứng qua đường chéo chính là những số phức, tức A = - (A*)t.

$\begin{array}{cc}A& \mid \begin{array}{cc}0& 2-\mathrm{j3}\\ -2-\mathrm{j3}& 0\end{array}\mid \end{array}$

Nếu ma trận vuông phức liên hợp có (A*) t. A = U = A. (A*)t thì ma trận A được gọi là ma trận đơn vị. Nếu ma trận đơn vị A với các phần tử là số thực được gọi là ma trận trực giao.

Bảng 1.1: Các dạng ma trận.

 Kí hiệu Dạng ma trận A = -AA = AtA = - AtA = A*A = - A* KhôngĐối xứng Xiên-đối xứngThựcHoàn toàn ảo
 Kí hiệu Dạng ma trận A = (A*)tA = - (A*)tAt A = U(A*)t A = U HermitianXiên- HermitianTrực giaoĐơn vị

cock cock cock cook cock cock Ock cock cock. cock cock cock penis penis dick dick penis cock?
Boy, looks like you have issue
gor
indeed
john
Hi
Albino
Hi
MentalHealth
taretts....lol or could suffer from pendejositus. that's spanglish for dumbasitus!
Jo
sensation such example
what is sensation and example
Fateha
Sensation is the process that allows our brains to take in information via our five senses, which can then be experienced and interpreted by the brain. Sensation occurs thanks to our five sensory systems: vision, hearing, taste, smell and touch.
john
For example, when touched by a soft feather, mechanoreceptors, which are sensory receptors in the skin register that the skin has been touched. That sensory information is then turned into neural information through a process called transduction. Next, the neural information travels down neural p
john
pathways to the appropriate part of the brain, wherein the sensations are perceived as the touch of a feather.
john
thanks john
Fateha
always a pleasure
john
five minutes to rest the best way to relief our stress.
Rhollee
I believe in the whole brain we need not think or even remember what events may have happened.I do think memories will always be stored like to think of the whole brain and heart.Guess that's another surgery..
I think what has occurred in our life also plays a role in our intellect.
Misty
Agreed.
Laura
the brain is a powerful tool for humans to handle, if you have full access to the brain, you can control pain or anything other emotions, for instance according to my research you can get pain when ever you want.
john
I agree with John full control of our brain we can be able to get thru anything ...
Sonia
@john you mean using our memory?
Cherrie
painful memories, yeah you can control them too, @cherrie depression is bunch of chemicals pain is too with pain receptors and motor nerves
john
wawoo amazing
john
thank you @sonia
john
if you want peace of mind.. simply don't fight with your thoughts!
omar
that's what I said brain controls everything, if we control the brain and thought process, things will be simplified, hell it will easy peasy life
john
hmmm
omar
why is learning important
you actually want an answer for this? :0
Sohaib
learning is important because without learning you wouldn't even be able to have asked that question, among a plethora of other reasons my friend.
iWasHacked
In case that wasn't a joke.
iWasHacked
do you think that the moon is holo as when nasa made any heavy impact on it's surface it is recorded that I was shook for a while.
Raman
Your been what? Behavior is not what you know it's how you behave yourself.. Learning is important to me as we learn more is gained in life's work.
Bart
Bart Gross that didnt even make sense dude.
iWasHacked
Ramen Prasad that I don't think has any correlation to the question.
iWasHacked
The fact that the President saw or not can be a fear although with children and events the president had on his mind can not be assertataind I'm in the thinking he knew of a eye event seeing than remembering I'd have to disagree..
Bart
I also disagree that the whole brain theory it's environment, DNA I believe that in some way the Memory travels through sensors nerve cells. thanks
Bart
Bart. you must be commenting on the wrong section...
iWasHacked
explain inteligents definition ,nature and characteristics
inteligents definition ,nature and characteristics
Faristha
inteligents definition ,nature and characteristics
Faristha
inteligents definition ,nature and characteristics
Faristha
you want intelligence definition and the nature of it and their characteristics
john
a teacher needs to understand and explain the behavior and activities of a learner both in class room and outside the classroom.establish how this can be achieved
how can a teacher understand the behaviour change of a students in class room
Doris
observation and conservation of treasured items
Crystal
how can a teacher observer the learners
Doris
Well I know a teacher who observes the overall mood of some of her students expecially ones who tend to yell at people who annoy them. She knows the steps to one of them about to go off and send the person outside for a chat (the person annoying them or the person about to go off).
another point please, How can a teacher observe the learners behavior in class
Doris
Just that observing patterns of behaviour
"observation and conservation of treasured items" All the way. Love it.
PJ
Do i need a lab and a instructor to carry out Psychology empirical research?
Do i need a lab and a instructor to carry out Psychology empirical research?q
Nikhil
Karen
i forgot to add, its Social Psychology Research
Nikhil
Depends on whether it's clinical OR a naturalistic research
Wayne
I was here before but got lost
Karen
hello
Sheik
hi
California
hiiiii all
I wanna be a young psychologist....
Sheik
Define perception of human nature and its significance
Psychology is important for different programs to study because of its ability to build students critical thinking and by so doing, improve their ability to sieve information, thereby improving overall learning outcomes
I have a question about sex addiction how can you tell someone would have sex with anything like child man woman old young even an animal I have a feeling the guy I'm with is someone that could do anything because he has no self control and he is a narcissist that also has a addition to meth..
How did you get to meet this guy
Wayne
hello may i ask How long have you been seeing this guy and what kind of signs were givin to come to this conclusion,? along with what you've shared on top here?
Jo
Facebook but we had the same friends but never met before but have been together for 5 years and its been hell I now have PTSD and now no intrest in doing what I love because he puts me down for doing so or makes me feel stupid for feeling the way I do about things
Alexis
been together for 5 years things were good at first he was really nice if I needed something he made sure to get it but once I moved I with him he started calling me names but once we got our own place he got abusive and pretty much had me as a prisoner in my own home
Alexis
also when he gets high he gets very turned on to the point he would masturbate several times a day only time we would have sex is if I was asleep or we would fight to the point of abusing each other then he would want to have sex but he also seemed to be very weird around kids
Alexis
but if sober he wanted nothing to do with him he would actually make me take my brother to my mother's room and not let him come out and play with his toys also I had my son asleep in his room with the TV and night light on seth said he was going to get a shower well the water was on I thought he
Alexis
it sounds like this guys is someone who can be very dangerous if under the influence. I highly recommend getting yourself and children away from him. He may seem like a nice guy but in my limited experience its better to play it safe not only for your own safety but especially for the children.
April
you gotta leave the relationship.. it sounds like it won't end so well and the fact that he's using meth; that alone is a very huge red flag
GL
I agree with GL Rocketeer
April
yes I do also but I want to think it's just the drugs but he still is very one sided and not caring of others feelings
Alexis
one thing I could share with u from individuals ive spoken wit is that a person who smokes or especially runs it becomes extremely "turned on" and or to the point were they'll masturbate or seek outside their relationship to satisfy their wants or needs. so honestly i dont think its a bad I idea...
Jo
fr them in this situation out of his sight,out of his mind!-
Jo
so yea id agree, pick up and flee! Good luck in your situation and may God protect u and yours during this time!
Jo
thanks you guys I've asked for advice from family but no one actually listened to me and just pushed it aside
Alexis
I also agree wit miss april, about methin around wit this guy, and already abused u already kno wat ones capable of. theroriticly if one day he looses his sack or been up for few days to were sleep deprivation begins to alter his jugment and emotions , possibilitys Are high, and could hurt someone
Jo
it is better to make a plan, get everything ready. Then make your move, if it is drug out over time and this guy still has access to you. He will try to convince you he will change and stop. He will agree with everything you say to make you believe him. it will be lies.
gestalt theory of learning with example sultan
Rohit
A friend of mine experiences homicidal thoughts. She doesn't believe she would act upon it, but it definitely happens more than the average person (which, most people at least in one point in their lives experience homicidal thoughts). She is a diagnosed sociopath.
Mostly I feel she would need a hobby or a purpose in life.. Finding determination to achieve something is tough for some people already, in your friend's condition it's even more challenging
Samrat
Friends and family are important support system.. She also has to reach out for help
Samrat
she is bored if the normal life, basically the drama is more around her I guess, so try to give her some puzzle, question her about, basically she is in chaotic thoughts, guess she has been watching some movies with homicidal content
john
A person who is bored with life, won't find anything to distract her. She needs someone who will listen to why she feels like this, and ask her what would make her happy, and work with what she answers
Jessamine
i agree with Samrat and John
Emily
@jessamine, sociopath behavior, if we keep on asking her questions about happy in the beginning of the treatment, she might be aggressive on happy
john
Didn't know she was aggressive
Jessamine
And Jessamine
Emily
Many things to take into account before actually doing something. psychologists have to analyze the person see how they act and respond
Jessamine
sociopathic aggression will be manipulation or conning, sociopaths can turn a person strength into weakness, and if piss off a sociopath the effect will be on others I mean surroundings will be effected or people, sometimes they might hurt themselves, they are incapable of love, I guess if we ask
john
questions or puzzles, manipulated people need some juice to brain and that juice is puzzles, or questions about her thoughts
john
yeah valid point miss @jessamine
john
good day can I ask something here?
Aiszel
what is the different of alters and multiple personality disorder? sorry I just want to know another answers about it *look down*
Aiszel
sorry @fermia you questions is quite confusing can you please repeat the questions
john
I suppose your question is what is the difference between alter personality disorder and multiple personality disorder? well both will be coming under DID disassociative identity disorder, in that alter personality is a system, and sequence they go through and multiple personality disorder is a
john
is a truma where the person will avoid the the pervious memories, with multiple personalities to over come the truma
john
am correct?
john
I can't speak upon that
Crystal
how can you calm yourself when in distress.. i cant simply calm my nerves with breathing.. negative result
Depending on the severity of said distress it depends on how extensive your feelings of anxiety & panic are. So the most universally applicable "safe method", would be to attempt to change subjects, as in doing something possibly unrelated to keep busy mentally
Miguel
If that's not sufficient enough aside from the plethora of other alternatives I'm sure you've probably searched online. I'd like to understand a little better how "difficult" things become? Hyperventilating? Feeling like a heartattack? Or just overwhelming apprehension, worry, etc?
Miguel
what kind of distress? And how do you get a negative result from breathing?
iWasHacked
i feel worry, hyperventilating.. i cant control my breathing sometimes.. my bad past shows up, then i stupidly doubting myself a lot.. cant even look ppl in the eye n lock myself in my room. i try the breathing methods on google but it doesn't work, maybe i need someone to help me. such a pussy
Nicki
how the hell did this conversation get on my messages
mauReeN
meesages
mauReeN
idk
Nicki
I'm still trying to figure out what this is exactly.
Jonah
I have problems with hyperventilating as well so I can understand, so usually when I'm having an episode like that I concentrate intently on one object (ex. the floor, its design, the details in it, how it feels under my hands, etc).
iWasHacked
The Last time I had an episode it got intense enough where my extremites went numb and tingly and I couldnt see straight(probably close to fainting) with an intense pressure in my head and I had focused on the way the floor felt under my hands(stimulation helps).
iWasHacked
even though I couldnt really feel much but I had calmed down. Another thing that could possibly help is reciting over and over in your head if you are able to "I'm ok, just calm down,everything is fine you are going to be ok, you are safe breathe in and then out" this is usally what i do on the mild
iWasHacked
cases when I am just starting but when the hyperventilating gets intense freaking out from the loss of feeling is i think normal but its important to try and stay calm and try some stimulation for distraction. Other times my mother has helped to calm me down a lot, she usually was the one to help me
iWasHacked
so maybe ask for somebody to talk to you in that situation to distract your mind. Another thing I find is its best to sit down on your knees on the floor because if you faint you dont want to hit your head and usually when hyperventilating I surely cannot stand up straight so its just easier
iWasHacked
and you'll have one less thing to worry about in that situation. I really hope one of these can help you, its a very stressful situation and I really hope you can find some peace. Just remember that because something works for somebody else does not mean it will work for you so don't get upset if
iWasHacked
these dont help you just keep looking for something that works for you. You could also try some meditation outside of your episodes and maybe incorporate the meditation when you hyperventilate? Good luck!💛
iWasHacked
sorry for so many comments it wouldnt let me put it all in one comment lmao. Hope it helps.
iWasHacked
12345
Graced
I feel anticipating the situation and knowing that it could've been worse, can calm you down.. you basically need to be okay with whatever life throws at you and think ahead
Samrat
how do you get someone to stop smoking marijuana
you dont ..They need to want it for them self no one can make anything they don't want..They need to learn to have self control to live with out it..
Barrera
anyone**
Barrera
I lost 32 lbs in 6 weeks because all I could do was dry heave. Don't judge someone when you know nothing about them or their struggles
Tara
first you need to pinpoint why they are using it. There may be a health motive behind it.. if so marijuana is the most natural form of any pain relief available. It has several health benefits depending on the user and purposes.
Even
I get someone from smoking marijuana when I look at the bad side of it
austin
you know guys marijuana is a med but all meds will be bad if youll use it exessively
facts
Anxiety is why I stopped
Gianni
First question is. Why are you wanting to get somebody to quit? It could be purely their choice even knowing the side affects and if that's so you cant really do anything. Last, how long has this person been smoking and why is he or she smoking?
iWasHacked
What are some usual cognitive behavioral therapy routines for people dealing with addiction? More specifically, a case of a 21 y.o man with a comorbidity of marijuana addiction, addiction to sexual fantasy & pleasure, and an addiction to food. If you guys could help, I would appreciate it :)
what does marijuana do in the body of a person?
austin
Got questions? Join the online conversation and get instant answers!