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.
Simetrik Matrisler ve İkinci Derece Kuvvet Fonksiyonları / Symmetric Matrices and Quadratic Functions
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Matrislerin Köşegenleştirilmesi / Diagonalization of Matrices
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Attila Aşkar
Prof. Dr.
Hello.
An important application of the matrix of eigenvalues and eigenvectors
We examined the diagonalized.
We see here a second application, however, these diagonals with diagonalization.Lines
Another type of problem but also about size.
To introduce the subject of the origin of the x and y independent
z is a function of the variables are defined.
This function would be three-dimensional space, x and y independently
this geometric body for variables that have two degrees of freedom.
This kind of happening surfaces in geometric objects.
Function of two variables but we address
When a force in a second degree function.
Of course there is x squared, y are square, there is a quadratic function in the xy'l term.
Function is very difficult to understand the nature of it when given with this structure.
In contrast, the term mixed xy'l terms
When available, easy to understand the nature.
If a and c plus valuable if a paraboloid z
will give the surface, it also paraboloid surface z
When we estimate we will secure equal ellipses obtained.
If a plus although valuable, valuable C minus,
Or, then it will be a hyperbolic paraboloid surface will die
We are experiencing a saddle surface that we call surface.
Here z is equal to the fixed value,
So when we cut the horizontal plane a positive value,
c we're getting less valuable because it is also hyperbolic.
Finally, c is zero valued here we also obtain a parabolic surface.
Has curvature in one aspect, we provide a non-surface curvature in the other direction.
So our aim is to get rid of the term B infection.
To do this, we will take advantage of the matrix.
But before crossing there that ellipse hyperbola
Let's go over the surface and the formation of a parabola.
When you cut the surface of a cone with a horizontal plane
As you can see a closed surface, it consists of an ellipse.
The slope of the cone in a plane with no
smaller than the inclination of the side surface
We cut with a plane
When we get a closed curve and an ellipse.
This cone slope, the same as the slope of the side faces
the estimation of the slope of a plane with an open surface achieve, we achieve an open curve.
But this open one branch curve, a parabola that we see happening here
If a fixed data such as z.
Similarly, the inclination of the side surface
We estimate an inclination to the plane that is larger than when the cone,
We get open but you get two branched curve curve.
They also hiperboller happening.
Now we will get them using matrices,
Although only two variables and then work easy,
and we can handle geometry, but more dependent problems
but they can understand that when the matrix.
Let our new name before a variable,
x x a, y x two say.
Wherein A, B, C had coefficients.
Had twice the number of B and C.
They also aim to define one by one.
A two combine to equal one or two of these men let it B,
x x occurs twice here because B is a two come from one or two,
so there will be an accumulation of a future than a two twice.
And of course, two of the squares of X future, it will occur with C.
These works show a more tidy if the coefficient matrix
A by product of XJ j x and i and j, we open one to two,
If we perform this sum, we get this function.
j i i a a a a time that is to say a frame of x j.
I have a jar two when x is a once or twice in this collection
x in the opposite when the two but i also when there is a two x two x j is happening.
Both also have the same value for accumulating here twice.
j iki i de
two x two times two is the time where a two x two,
We see that consists of two x squared.
There are already four terms, I can be one or two of them each
There are four terms in one and two because it comes against a j,
Four-term work emerges in this way.
The reason being, of course, a three with a two and two equal from one
x from one of two x two with x, x a x two also due to uneven.
We can show them with the fact that the matrix
Let's show with what is shown here with the coefficient matrix multiplication.
Matrix middle, right in the column vector x x two,
still left in the row vector x x two.
This time we made a product that is completely secondary here
We find the function.
Considering the general that
n-variable function is displayed in the same way.
I n terms of collection and expenditure only up to j.
In the middle, the right column vector in this component have
There vector x, left the line because they have to be
I get hit by a column vector column vector of the matrix.
This column vector you get hit by a number in this row vector,
We say that in our numbers.
x is an x two variables, we obtain a function of time.
n plus one dimensional surface in space, of course, these hands touching our eyes
not a surface we can see because we live in three-dimensional space,
but we can grasp them with our thoughts.
The process is very easy to do, our goal this complicated terms
before we see more clearly in a x x destroy binary terms.
This will make the course a matrix transformation.
He also said let the transformation matrix; Let's work with vector x instead of y vector,
that when x is obtained from y we provide a conversion.
When we put it x y transpose transpose times Q
It will be overturned.
yt hit Qt.
It stands in the middle, y Qy.
Disconnect the middle of the triple matrix multiplication,
As you can see again it becomes a matrix transformation.
Because the symmetric matrix Q t Q the opposite happens,
Q if the eigenvalues of A,
but we can get the size of an optimized eigenvalues.
We did not do it for no reason, it is always something that can be done,
After you have found any vector happens if you divide the length of vectors.
This time we knew it was a diagonal matrix multiplication.
A matrix for this transformation it can always to be symmetrical.
Therefore, this product because it's the end of the middle lamp diagonal matrix,
now remains only checkered terms of this product, there is no mixed terms.
And then we see the use of the diagonal matrix,
For this product to be inert in the eigenvalues on the diagonal
we obtain the structure of the zoom function.
The initial variable was the variable x, x, x two,
When we returned to xn, but as spontaneous natural matrix y
In a way, it has come to terms year on the principal diagonal structure is structure.
Let's make a visual statement that in three dimensions.
Here again we eliminate matrix.
Right column vector x, x vector are left on the line,
The challenge of this product as well as work on the diagonal checkered terms
as a matrix for a two by two symmetrically from the term
A one to two twice this coming because there are two equal x x iki'l term.
a three, a x a x UC to be equal to three bir'l
The term comes twice likewise de x two x UC.
This is obvious when we made the transformation remaining
The term is diagonal matrix, and z
QT AQ is a diagonal matrix where the structure
diagonal matrix is made here, and the lamp on a diagonal,
two lambda lambda has the eigenvalues as three.
This time we made the product because the terms are zero except the diagonal,
The term for the longer term mixed medley came from here
will not only x, y, a plaid, plaid two years,
quadratic terms will be three years.
If a lambda lambda two, three lambda value if all of them plus you an ellipsoid
The structure will be generalized, some pros, some cons of the hyperboloid,
hyperbolic paraboloid or zero will be the generalization
If you have been in that direction will also be parabolic sections.
Let's start with a concrete examples now.
In fact, we can do a combination of three samples of three.
In the first example of x squared plus y squared has xy
coefficients of a mixed term, the second sample mixed coefficient,
The term coefficient of the mixed four, two in the third example.
In general, if we look at all of them as we can expect the same quality,
but we will see that they will always surface composed of different nature.
In the first example of x squared plus y squared diagonal for a
on a future, because it comes from the checkered diagonal terms.
Than half of a diagonal of a term other than two,
Half of this will be from a two to a symmetrical structure.
Half, we distribute, including a half here in their totality
one half times two plus one half times x x x
two x will produce an assembled gathering here xy'l the term.
In the second example still has a same structure on a diagonal,
but in terms other than the two diagonal it comes as two.
A recent example comes from a well outside of distributing these two half the diagonal again.
As you can see these qualities as they like each other, but against incoming
We will see that the matrix and especially when we account for different eigenvalues.
In the first example I now give the final accounts before entering these details.
Divided by three first two eigenvalues will be either a split second.
Indeed, if we look at the total lamp on a diagonal
lambda would be a plus two.
As you can see a plus giving a two, a three split iki'yl to
divided by two to four divided by two that have collected two going, providing a plus.
Multiplied by four, divided by three it gives the indeed here
If we look at a minus one divided by four, three divided by four gives,
We see that such provision.
Here we find that a three and cons of the eigenvalues,
that nature is completely different, because in the first lamp also positive.
This though, it's important sign of it.
But of course there are a number of important nature,
In a qualitative status of the lamp and lamp will determine the two signs.
Plus valuable in both the first example, one plus one in the second example
less valuable, roots dating two and zero in the third example,
we can make it again.
We are also on the trail of a total diagonal matrix on a plus two.
Really the sum of two positive roots is zero.
The product of the roots gave determinants.
A negative one, zero, double zero, giving zero.
Although we have come here, see a total of two still on the diagonal,
minus three gives an Two.
But when we calculate the determinant gives a minus three minus four.
Indeed, a lambda lambda gives two minus three.
We saw this in the previous sub-section of the matrix
We understand in terms of invariants.
So when we go to the construction of the first prime prime function
Make a split two years, two years is a square divided by two plus three frames.
The second three times a year y squared minus two frames,
A function where y is the fourth frame.
No two years.
Now we see the difference in quality immediately.
Numbers are not important here.
This one half was unchangeable nature if not in terms of 1000,
Of course it would change the shape,
plus plus plus or minus here, where the plus and zero.
These opposing surfaces are roughly as follows: z is equal to x,
square denominated first variable
When the square terms plus a second variable,
z is the time we cut, we get an ellipse with hard surfaces.
Indeed z say you have a job that would be an ellipse,
The semi-ellipse axis lengths in particular here,
We find that the root of them easily reversed.
When we look at the pros cons of this situation
called elliptic hyperboloid, if this kind of a curve defined as the surface.
Next there is a parabola an upward direction, a downward direction of the two year
We see that there is a parabola, and z equals the time constant hiperboller we put out.
For example, say you have a z gives it a hyperbole.
Latest surface so that the axes
It is provided so as to introduce attributes.
There is a cylindrical surface for two years, this is how a cylinder,
You gonna draw a parabola to draw a parabola such as z equals x squared
When you shift this year and two independent
ötelenmiş this means for the way in that direction would be.
Such a paper to take your work to a parabola buks
parabolic cylindrical surface will be obtained.
But did not mention the Q where Q
If you remember our previous example, cosine
cosine sine alpha alpha alpha alpha minus sine
meant opposing a rotation in the plane, that is to say cosine alpha
The root surface is 45 degrees turned a divide that by two.
For example, here we have tried to reflect a little, a surface that turned 45 degrees.
It is also a 45-degree surface could reflect back.
Another example let three variables that time.
x squared term, y squared terms,
z squared term as well as xy'l have yz'l and xz'l terms.
Again, this can be written in terms of a matrix.
Seven on the diagonal coefficients of the matrix,
ten and seven will be, except that the first diagonal terms
As you can see here minus two minus the sum of the two will give it.
Other terms is a one wherein
AI is half on one side for what we call the half on one side
still less because it is a minus in front of the xz'n
negative values that reflect where we are.
This problem, we find the eigenvalues of this matrix,
It is given on page 198.
12 and six eigenvalues, the eigenvalues are also repeated twice.
So no more a feature of this example, I just re-çözmeyel eigenvalue problem,
Consider the results of earlier I said we solve a problem of this new
on variables
As before, we define x Q.
say y, y components, y1, y2 diagonal terms
which consists of a square matrix Y1 here for six times,
plus six times square Y2, Y3, plus two times the square,
all of them work as he hit six two, six, six of them is formed.
This is a high-dimensional surface,
that is, a kind of four-dimensional ellipsoidal surface is a surface that
We decided that because we did not draw the four dimensions, but three
We find it reflects the experience gained from dimensional surface.
See here, in particular the function of what is now
We determine the properties of the surface can not remove.
This is what we think that's y he opposed revenue Q.
we have defined as occurring when x
Q matrix, eigenvalues, one size,
we created this unit from a height reduced to eigenvalues
orthogonal matrix, the matrix is orthonormal.
This represents an opposite rotation in three dimensions.
We have seen here kosünüs sine alpha minus alpha, alpha sinus,
kosünüs alpha while opposing a rotation in two dimensions,
wherein Q matrix comes opposite to a rotation in three dimensions.
To illustrate this with a concrete example with such good matrix
i Q i of vector that we have with them çarpsak multiplied by Q
When we get a vector of length unit again.
How will it be?
Here is a bring a zero, put a zero horizontal turning them stand out.
Of course, zero reset, it remains for them not to be hit by the second and third column.
Only one half of a front, emerges as a zero minus.
Even if we multiply by j would remove take the second column, we're getting away.
The third column by k, so as ijk
We're going to pull the orthogonal Cartesian coordinates of each.
This rotating vectors,
We know that changes in the size of the rotated vector.
They base e1, e2 base, because let's e3 base before
on page two hundred and seven theorem we have seen,
The new vector obtained at the end of this rotation again
He says the terms will remain the same, between ijk,
ijk was ninety degrees to the angles between them.
So the base e1, e2 base, the angles between e3 base
It will again be ninety degrees, will then be perpendicular to each other.
Height will stay in one.
Let's look at providing that.
e1 and e2 we hit the base of the base, see,
plus six plus one divided by the root minus one divided by zero
a negative one over six roots because the roots gives zero with six.
As you can see again with e1 e3 çarpsak
see here in a split twelve turns denominator, a cross hair, but important.
Here comes the dual.
This gives us the same values.
Baksak the length of these vectors
A second frame of the first component divided by two plus
divided by the square of the two components; a total.
Here we look, a division of three, one divided by three,
a split to give the squares of the three components.
When we look at here as four plus one plus one,
ie four plus one plus one; six.
Therefore, they are also coming to collect six coming in the denominator for each other.
We see that the same length of stay.
So we finish this chapter.
Your work, apply what you have learned it
To özleştir,
I would encourage you to resolve these examples in order to internalize.
Some of these non-symmetric matrices,
A portion of the symmetric matrices.
Roots, which could be easily found because either
each has had many zeros or repeated numbers.
If no one challenges the duality of what has already doubled.
These are the challenge of the theorem we found earlier
You will be able to internalize.
Again, as we give a summary at the end of each chapter.
You understand this concept, where you have missing,
a summary of which you understand well.
Thus, you will get a chance to test yourself
and it's the result of a combination tidy.
Bye now.
Now nearing the end of this course, we will see matrix functions.
Just as the frame of a variable x, cube,
If we can get n'nc forces, we can achieve if we gather strength functions.
Sine, cosine, as bases function is not
that can not be represented by a finite number of forces such as the logarithm function,
There are functions that can be represented with infinite Taylor series.
Likewise, find that the functions of matrix and their possible
properties used in many applications.
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