Doğrusal cebir ikili dizinin ikincisi olan bu ders birinci derste verilen temel bilgilerin üzerine eklemeler yapılarak tamamen matris işlemleri ve uygulamalarını kapsamaktadır. Cebirsel denklem sistemleri, sonuçların tekilliği ve var olup olmadığı, determinantlar ve onların doğal olarak nasıl oluştuğu, öz değer problemleri ve onların matris fonksiyonlarına uygulanışı vb. konulara derste değinilmektedir. Ders gerçek yaşamdan gelen uygulamaları da tanıtmaya önem veren “içerikli yaklaşımla” tasarlanmıştır. Bölümler: Bölüm 1: Doğrusal Cebir I'in Özeti Bölüm 2: Kare Matrislerde Determinant Bölüm 3: Kare Matrislerin Tersi Bölüm 4: Kare Matrislerde Özdeğer Sorunu Bölüm 5: Matrislerin Köşegenleştirilmesi Bölüm 6: Matris Fonksiyonları Bölüm 7: Matrislerle Diferansiyel Denklem Takımları ----------- This second of the sequence of two courses builds on the fundamentals of the first course, is entirely on matrix algebra and applications. Specifically, the studies include systems of algebraic equations including the existence and uniqueness of solutions, determinants and how they arise naturally, eigenvalue problems with their applications to diagonalization and matrix functions. The course is designed in the same spirit as the first one with a “content based” emphasis, answering the “why” and “where“ of the topics, as much as the traditional “what” and “how” leading to “definitions” and “proofs”. Chapters: Chapter 1: Summary of Linear Algebra I Chapter 2: Determinant Chapter 3: Inverse of Square Matrices Chapter 4: Eigenvalue Problem in Square Matrices Chapter 5: Diagonalization of Matrices Chapter 6: Matrix Functions Chapter 7: Matrices and Systems of Differential Equations ----------- Kaynak: Attila Aşkar, “Doğrusal cebir”. Bu kitap dört ciltlik dizinin üçüncü cildidir. Dizinin diğer kitapları Cilt 1 “Tek değişkenli fonksiyonlarda türev ve entegral”, Cilt 2: "Çok değişkenli fonksiyonlarda türev ve entegral" ve Cilt 4: “Diferansiyel denklemler” dir. Source: Attila Aşkar, Linear Algebra, Volume 3 of the set of Vol1: Calculus of Single Variable Functions, Volume 2: Calculus of Multivariable Functions and Volume 4: Differential Equations.
Çözümlü Problemler: Üç Uygulama / Solved Problems: Three Applications
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Из курса от партнера Koç University
Doğrusal Cebir II: Kare Matrisler, Hesaplama Yöntemleri ve Uygulamalar / Linear Algebra II: Square Matrices, Calculation Methods and Applications
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Matris Fonksiyonları / Matrix Functions
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Attila AşkarProf. Dr.
Matematik Bölümü (Department of Mathematics)
[BOŞ_SES] Hello.
Matrix functions related issue that I hope you'll find interesting from our end,
three types of application will deal with three problems.
It's called the transition matrix to one of them.
Transition matrix shows the transition from one position to another position.
Or, in the economy of rural sociology in this instance
Topics examined the migration to the cities.
Long transition from one location to another, possibly theory
the possibility of the period.
Because the transfer of energy from one location in Physics
from place to place energy guarded, it protected them show events.
Producing a laser beam
As you're sending a beam of laser light is reflected from a mirror,
Coming back, it's going to grow each time, it comes somewhere in stable condition.
They are a protected energy, protected population,
There are features such as protected probability.
This shows that the sum of a matrix column to come equal.
These transition matrix "Transition Matrix," it said.
Again, this one is similar preys hunted problems and issues.
It shows the confrontation between the two rivals.
Adjustment of the number of rabbits will we do with the nature of the fox.
But it's the equivalent of problems in the economy,
on the issues of the problems of competition between-species evolution.
You may have heard the name is also called the Fibonacci sequence.
A sequence used in the aesthetic ratios tested this golden ratio.
It is very interesting as well, jumping to 1300 years ago
Interesting subjects of a scientist in Italy,
breeding of live plants in nature.
In these very different rates, in places, also used in the field.
When it's done looking at the growth of the rabbit.
How could he look at her numbers to urea.
All of these issues are described as follows in fact,
mathematical side; We are the first in position a short cons,
We can say the current time.
This is an evolution of events, interactions k'yınc the result of passing the time.
k'yınc short time after this time instead of a minus
If you put k k plus passing first time.
In this way, evolution occurs.
Initially we have a given initial value.
Suppose we show the vector x0 that.
This is turning into a vector x1 x0 of the vector,
that provides an evolution, provides development, lets change.
position x1 x1 x2 when it is in this time
x1 coming and going in the time frame that you insert x0.
If you repeat this short time k'yınc position
A force of k'yınc comes to original position.
This is an application of matrix functions.
We will see a second application in subsequent sections.
In solving differential equations.
No need to know Masia scare equations, matrices problem.
Now I think that is how their formula.
Let's say; k'yinc years of population in rural areas
ie the ratio of a country, or even in a city we can be certain.
It can also in some bacteria, can also financial issues,
You can also energy issues.
But the more they settle this with a concrete example as population, population migration.
I would also interesting,
We already live in the spread of the population in the 1950s in Turkey,
with the distribution between rural town, we are a lot different now.
Let's also tbsp population ratio in urban areas.
So call those in rural areas XKR
We say to those in the city tbsp we found this time.
But where I come from, this did not come without rhyme or reason.
Suppose one in each year from a previous position.
As every year, I think that dynamic.
The figures are a bit unrealistic that he chose.
Four percent of the city from the countryside,
I think he went to the city's four percent of the population in rural areas.
Of course a portion of the people in the city
retired happening, he wants to develop its business in the region where he was born.
There are also a return to the rural areas in the city.
This is but a smaller proportion, so they both get back to rural areas.
Now we will see it in this formula
number of rural areas in this corner of the matrix
The proportion of the population to the city.
Because gone four percent.
When we look at those in this city
Two percent goes back to the countryside, ninety-eight percent stays in the city.
So the remaining ninety-six percent rate in the rural areas of this country,
four percent going to the city.
City continues to remain in those ninety-eight percent of the city,
So going back to them.
In this way, we create a matrix.
Vectors in the matrix of two-dimensional vectors,
we show here that Xk- 1 column vectors with the XKR's happening.
If we say that we see in this matrix equation it comes as a structure.
You stood with Xk- 1 gives the XK.
Now he was coming down like this in Turkey and fifty years
sixty years in such a population exchange.
Initially percent of the population in rural
Suppose that the eighties, was the case in the 1950s.
And of course Eighty percent of the city total
Twenty percent were in the city when the rural population.
So we need a vector x0.
This will multiply the forces of k'yınc we saw that x0'l here.
x0 means that zero, eight, zero, there will be twenty.
Now let's throw this in mathematics.
k = 0 at the beginning, so we choose sixty seventy years previous status.
This vector x0.
This means that the problems we have as our mathematical
Then calculate the k'yınc forces.
given x0.
A given.
Here.
We will calculate the k'yınc forces.
How do we calculate e?
We will start by finding the eigenvalues.
Here we write the matrix.
We marshalled minus lambda on the diagonal.
This will be a function of the secondary lambda.
We find that when we reset the lamp in a two roots and two lambda.
We find the vector that they opposed.
I did not do this interim.
That will put one lambda.
See you put one here lambda lambda a time, one turns out.
Here minus zero, minus four here will be zero, zero, four will be released,
zero times x.
zero, will be with twice.
To find the x and y, you will find also a y x.
Similarly, if you figure it out here lambda two figures in
We select the appropriate work out well, because we have already.
Here two values of lambda
We find that the zero ninety-four.
The corresponding eigenvectors here.
This would form the core matrix based vectors Q columns.
This Q-1 of the matrix Q we need to find.
For that, you need a determinant.
We know that two binary extraction in the inverse of a matrix.
Multiply that out if indeed these two units also provide matrix
You can.
To find the strength to say again k'yınc
revealed not only function where the k'yınc force,
For any function that we saw in a previous problem.
This function on a diagonal and lambda lambda
We are writing calculated values in two.
Lambda bir birdies.
K'yınc force one.
Two in the lamp, I would not write it as numeric, two of k'yınc force lambda.
K'yınc obviously going to be a force of one.
Q. I've written have brought to the left side.
Q-1, we wrote, we bring the right.
Look here for a three, there are three factors of a split.
This product is not a difficult thing if we do this two matrix multiplication.
We obtained the following situation.
If you wish, we can provide a k a
I need to give him time to come.
It was already in the matrix.
If you really make this product, if you put the two values in this light,
As a lambda is already a force as a whole.
Indeed, we provide the matrix.
This is a checksum.
When passed a long time now, now when k forever
but see the two will go towards lambda value than it was a number of small,
As you increase k it will shrink because of two lambda.
Therefore it will not change a lamp and one was in the end
Lambda deuces in the big value to be dropped,
of course we can not go on forever, but it will be close.
It will be the limit of infinite we say forever.
The resulting matrix as it happens.
This matrix in stable condition in the long term,
The strength of the latter, the fact that infinitieth force, so that
You can not otherwise account of the need to multiply their time with endless.
We are writing the value of the long-term here we multiply it by x0.
So the determined period, find the value in the long term
We stood for the initial vector x0 to the matrix,
the result is as follows: It is going two against each other.
So the population in the rural population in a happening city it is.
If you also consider that Turkey is going down.
Turkey is not simple, because of industrialization in all countries
It was the influx from rural areas to cities.
In our example initially we made eighty percent of rural nüfustayk,
Twenty percent of the rural population against the city that was in town in a four person
when people see is down to one person against two people in the city you see.
For example, if you choose such k of sixty,
From the 1940s to the year 2000 there, 2010s to the year
It was indeed the population as a result of this migration to accumulate near the population.
It can be used in a further example of this kind of problem.
I told the Economic Problem,
Reaching saturation of laser physics, such as the formation of a laser beam
It is used in problem.
Used in logistic problems, namely industrial engineering problems
as used in many places such as action research direction.
In the second example we will call the transport matrix,
roughly the same kinds of problems will recall the previous
every moment was becoming a sum of columns of this matrix.
See here but we split three to three, where three,
we split three to one.
Initially we were given to collect a column here and here
K'yınc to the beginning of the force were finding that we had hit the place.
Here again similar problems.
Now this is called also hunted prey to this problem.
Now again to make concrete,
They also have applications in other areas such as economy and so on.
Whether foxes and rabbits.
Of course, they hunt foxes and rabbits.
From one side of the wire fox population is increasing due to the growth,
thus increasing the population of rabbit reproduction.
But the population of foxes and rabbits avladık of declining,
foxes and rabbits also need to eat in order to survive.
You can also see this competition as economic opponents Or, energy
They position between competing with each other in problem
It would have similar problems.
Now here it will be a transformation.
It made a definition; XKR population of foxes,
also tbsp
Whether the population of rabbits.
Now, let's start with the population of rabbits here before.
Increasing the population of breeding rabbits.
So this parameter is selected Where'd you say them appropriately.
You will see why it is appropriate.
These will be different unless you change.
If you think of the event, if the little fox rabbit population will devour rabbits
but with the result that when the fox rabbits devour rabbits
so the food will not die will begin in rabbits.
One case is that no two people.
If the population of foxes and rabbits plenty to eat to eat them
They will not finish, will continue to increase in the rabbit population.
Reciprocating the food will always, therefore, will increase in the fox population.
Two populations can be increased dramatically as well.
There is also a steady state of equilibrium.
When you get a suitable ratio goes to a stable equilibrium.
This work will go to the stable distribution of population.
As parameter with a bit unrealistic, the fox
increasing population, increasing reproductive.
But foxes are not increasing as much as rabbits.
Here also show an increase in the population of rabbits.
But the declining population of a little rabbit, fox to devour it.
See, here comes the minus sign him.
Fox rabbit food even more plentiful.
For him this is proportional to the number of foxes.
When we look at the fox reproductive population is growing, but less of them
The fox population is increasing by the rabbits.
See here a number greater than one, where a small number but one
The fox population is growing in the seven rabbits.
If therefore there are a lot more because they can eat more rabbit
fox feeding so many natural deaths,
Deaths from hunger because it is increasing fox population decreases.
See both the positive signs here.
This naturally increases with urea,
they find food to eat because of this increase in rabbits.
Here's the minus and plus signs.
This second issue of reproductive rate of eighteen split six rabbits.
But foxes eat rabbits have decreased since.
If we wrote a matrix,
See the coefficients of the matrix, let's write sixteen of taking out a split.
ten, three, four and eighteen minus happening.
This yıldayk k minus the first case,
this matrix so moving that information when we came to k'yınc year.
This is called transfer matrix.
Transition matrix, but there is actually a transfer matrix columns
a sum, wherein the sum of the column does not.
See here for six-sixteenths of a twenty-one-sixteenth the other hand,
side it is called more generally transport matrix.
Transfer Matrix said.
This is a special state transition matrix, a sum of the column.
Now, this is what we will be in k'yınc state that,
What k'yınc year, and what will happen in the long term, if we say that we know,
likewise this dynamic zero years hence,
which shows the dynamics of the matrix, you can find people that first year.
If you know the population in the first year, second year, the population of this again
Relocation due to transport matrix with the matrix,
hence the transfer matrix you find x2.
But once it comes x1 x0 frame of when you put it here.
If you do this repeatedly in a short time it comes k'yınc force.
If you look at the long-term need to calculate very large values of k.
Now the first job will be to find the eigenvalues of this matrix.
Because you will remember it or you or diagonalization.Lines
You'll find the Cayley-Hamilton theorem using eigenvalues.
Let's say we think because it is a simple process via diagonalization.
Here we are writing this one split matrix including six out of ten.
We will write on the diagonal minus minus lambda lambda but this one over sixteen
In order to receive notice of lambda divided by a trailing sixteen,
Did we say sixteen times a ma lambda converge here.
So this is a detail, you would put six of them inside if you like,
You say here we find the value of lambda me here.
In contrast to the 16 divisions in this direction are lambda opposing this on the outside.
This is an account.
Simplification effort.
We find a Lambda, Lambda one.
The three also divide the two Lambda smaller than we find the number four.
Here we have put this or lambda,
We find them self in opposing vectors.
Again, the same way we do in the longitudinal self-vectors.
We put in the first column and the second column.
Here there is a need Q-1, on the contrary.
You need to find the inverse determinants.
Here comes here two binary inverse of a matrix.
If you do this you will see it multiplied.
This process
After finishing on the diagonal lambda
One of the kth strength, we put the two in light of the k-th power.
Left Q, Q-1 side.
Here outside there was a split of a four factor.
This challenge results obtained by this multiplication.
This is a general solution to this matrix.
Of course, a lamp, it was a.
Thus, the k-th there will be a one lambda force.
See coming here as two.
Two 0.75 lambda has three split was four.
Here we find values.
Two 0.75 lambda for that three-quarters.
We put two lamps are also here.
So, we got into the k-th power.
If we go to the great value of the k-th power, two 0.75 lambda,
namely because it is smaller than the longitudinal force shrinks value accrues.
0.75 in 0.75 times, will shrink to a more 0,75'l size.
So, when you go for a long period, the lamp will fall on both sides of the kth forces.
And these numbers will only stay back.
This is the infinite, so infinite time.
Of course, when in reality it will not be forever, but for large values of k it quite
value will be close.
These take place, we replace,
We stood x reset, the initial x
Next, we determine the zero zero
but initial x and y is zero zero,
for example equal to two may be selected population it can be selected in a slightly different population.
In the long run, when we multiply it by x year zero reset,
We find three times y times x minus two zero zero.
So x number of foxes initially zero.
The number of rabbits three y is zero.
This is supposed to be the beginning of a great number of rabbits
plus valuable you go and get a valuable longitudinal plus population.
If you go to a valuable minus population already depleted populations.
Rabbits and foxes are also depleted.
Here we show this process at work.
In the case of the infinite, that the situation in the big time,
originally derived from the value that we arrive at the stable state of equilibrium.
As you can see in stable equilibrium,
the fox population rabbit population had to be doubled.
Going rather not have, to arrive here.
And it's also the beginning of a factor as follows.
While initially the number of foxes x zero
rabbit year is zero
while this year the number of zero divided by zero rate of x,
We see that there is a two in the long term.
Now depletion in this model generation,
Let us just now said that the coefficient changing a bit.
See here for ten compared to the previous one,
three, four and minus 18, maybe you can keep them in mind.
Here are ten, but had three,
some roots here with minor changes,
eigenvalues, lambda divided by seven to six,
The lamp also comes two four-three split,
lambda is also smaller than that; and two smaller than lambda.
We found them before, we find three pages ago
If we place in the overall solution, as you can see here six split seven divide,
pardon the six split seven kth force has three split four k-th power.
Both of these numbers because it is smaller than when you go to long-term,
both go to zero.
Hence the value is equal to zero in the very long term matrix.
This means that, in the long run the foxes will devour rabbits,
rabbit will not, I will go to zero,
Because the population of foxes in their remaining food will go to zero.
In fact, here for humanity,
A mathematical model worth considering for environmental problems.
So,
it would be better to live harmoniously in peace, even though competition.
In a third case, we can say that the population explosion,
The completion of the number of rabbits eat smaller
If you hold and you select the appropriate growth rates, the coefficients
One of the core values greater than one turns, the other also comes less than one.
Let's place instead of them.
Of course k grows smaller than the value will go to zero.
k grows, which is larger than the value will go to infinity.
So if we account any k value,
When one half of the k-th power goes to zero,
For it will not be exactly zero but will go to zero,
See if we reduce them under common factor remains divided by seven.
Here, there.
Others went to zero.
Seven of gold divided by the k-th power in this way will be a growing number.
Here we leave this, the number of k grows fox will increase,
The number of rabbits will increase.
This is something that can be called population explosion.
If you think of it in other areas, the population of the world
consider, consider other economic models,
consider energy problems.
This population explosion, you can see the explosion of the population.
That same model parameters change,
It can give very different results.
The third problem we call this the Fibonacci sequence.
Fibonacci have seen that, they are also very beautiful pictures,
EnterNet can look and go.
There is a nucleus of a sunflower in the middle, there are two three core around him or something,
in this way the growth of the largest circle that has more numbers.
As the leaves, as it is possible to see many things in nature.
In economics, you can always find examples of this.
Imagine something like this, you get a rabbit set.
That cluster, cluster telling the mathematical sense.
They themselves are the first generation in one.
These groups themselves, such as to produce a second population but
When we arrived at the place, it made people who will produce, will be a themselves.
You can also assume that the first group had died.
And in this way the two, picking up the two numbers
that will produce two, three of whom will have to produce.
So here it is done as the Fibonacci sequence,
If you start it you collect a combined his own.
You collect one to combine the two.
Birla add up two and three.
This place has gone a very regular basis.
Now you collect two plus three is five.
You collect three to five to eight.
Five of the eight 13 you collect.
As you can see the numbers continue to rise rapidly.
Now let's write it as a matrix problem.
where c is a zero.
When we passed a car Or, in any case,
You collect the following number with a previous count.
For example, where you collect two plus three, you'll find five.
Five-k plus the first position.
How do we find five, we find by adding k minus the number of the first to k-th.
So, we get such an evolution equation.
Putting it on the heads of factors, such as this one, not one,
Putting is a different thing, the development of the country
You can get up development work in nature ..
Here is the most simple when it comes to selecting the Fibonacci sequence.
Let's say a vector ck -1 here.
C k plus 1 times vector c k to say.
If you choose again here again as follows.
How diyosunuz here?
You receive him.
You receive one himself, you are adding to the previous one.
This is happening to a problem identified by matrix.
Symmetric matrix has zero on the diagonal one.
While initially 1.1 x0, you're doing here.
2 gives.
When you do one of these two gives three.
See if you place here longitudinal vectors
You can produce these numbers.
(1.1), plus a second output when you hit a good turn horizontal.
When you do now because you put the CK-2.1,
who will put a reverse 2.1.
There are 2, where 1 while hitting horizontal turn 2 and 1, you find 3.
Now if you put 2.3, 2 plus 3, you find 5.
That matrix is producing it for us.
This last line also gives an identity as equals ck ck.
Now here if CK-1, CK remain invested when you hit.
Wherein c gives k.
So this last row of the matrix gives the identity.
Again, we do it in the same way, in the future k'yınc force.
I do not want to take it to the end.
Here I give you the details.
This equates to examine the matrix of the formation of the Fibonacci sequence.
Here I give you the details.
Again Q'yl the Q -1'l the roots, the eigenvalues on the diagonal
Type able to get by with this cross to k'yınc forces.
An interesting surprise of even an extension of the Fibonacci sequence.
This is called the golden ratio and in architecture, sculpture,
The images used since ancient Greek
used the golden ratio shows the relationship between these dimensions.
Leonardo Da Vinci's has a beautiful picture of a human image.
Never in such an unexpected application.
I'm a little bit of how the subject of interesting applications
We have examples that might be wide horizon, I thought.
In this example I gave to him.
We finished our subject.
So that was the subject matrix functions.
As a summary each time, giving the main issues.
We have a lot of issues.
No sub-topics.
To the zero power unit becomes the matrix.
Left or right because it is a square matrix k -1'inc
We find strength to strength and hit the k'yınc equal turns.
You know, the multiplication of each matrix,
B is multiplied by A in general is not equal to the multiplication of A with B.
But because the matrix does not matter here because it is multiplied in the queue.
Because it is the square matrix.
After calculating the strength of the matrix here now
We can obtain the power function.
If such an infinite number of base functions that even the bases
i'l to çarpılmış, sinus, many functions such as konsinüs, logarithms etc.
such functions may be calculated.
If these infinite series converges,
If the digital function converges on a single variable x, which matrix
functions are also converging and saw them in the calculation.
If the eigenvalues are the lambda
f (A) of the eigenvalues have been calculated with the same function.
That the functions of the Lambda,
If we see that the essence remains the same vector.
We saw it advancing various applications.
The matrix of the core values that change
We have seen under similar transformation.
We saw invariant of the matrix and thereby advancing our issues,
We have identified two methods to account for matrix functions.
One so that Cayley Hamilton theorem
n -1'inc force only until any function
The show can be ended with a knock account.
Hence, even if x cinsind from an infinite series, for example, consider sinus x,
to think about the x in terms of x terms of a numerical
while infinite series, comes a finite series for the matrix.
This is awesome.
Surprisingly you can think of it as a result.
Using this feature are the many unknown factors,
this n one account unknown coefficients comes to such a matrix solution.
The second way is how a matrix Q -1'l to the left,
Q'yl right to hit the Q matrix of the essence
including vectors, we were getting a diagonal matrix.
Or, vice versa time we do more
When we look instead at the genelletilmiş f (A) 's conflicts.
In contrast, the f (A), this method turns out to be calculated.
Of course, just because it can not possibly summarize it possible to grasp the subject.
But also use this summary, what you understand,
it is a good opportunity to test to see what you have left incomplete.
Done a good opportunity to review those.
Lastly, this migration problem, or balance between competing
exhaustion or both of the explosion that forever to arrive and the Fibonacci sequence
We saw it as the natural forces of the matrix with the transport matrix.
Bye now.
We will then switch to another application: Differential equations.
This suggests that scare.
The only thing to do, bases the calculation of the strength of a matrix.
The next session in our new
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