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.
Hermite Matrislerinde Öz Değer ve Öz Vektörler / Eigenvalues and Eigenvectors of Hermitian Matrices
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From the course by Koç University
Doğrusal Cebir II: Kare Matrisler, Hesaplama Yöntemleri ve Uygulamalar / Linear Algebra II: Square Matrices, Calculation Methods and Applications
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From the lesson
Kare Matrislerde Özdeğer Sorunu / Eigenvalue Problem in Square Matrices
Meet the Instructors
Attila AşkarProf. Dr.
Matematik Bölümü (Department of Mathematics)
[BOŞ_SES] Now
We see eigenvalues and eigenvectors of the symmetric matrix.
We want to define the Hermite matrix.
Hermit matrix means that: Elements of this complex valued
but not necessarily all of them can be complex valued complex
found valuable items, it's complicated
we take away the conjugate transpose matrix of time equal to itself,
We will now see already now an example, we say Hermit matrix to the matrix.
Hermit, it comes from the name of Charles Hermite,
this really made a significant contribution and have lived on this date 19.
An important mathematicians of the century.
Real matrix is already No.
It means taking the conjugate
it means, that the conjugate is itself the real thing numbered,
so overall than symmetric matrices to the Hermit,
A special case is going real symmetric matrices.
It turns out the real eigenvalues in the matrix,
going to each other and repeated eigenvectors
Even if we can find independent eigenvector eigenvectors,
So here we find the same features symmetric matrix.
The only difference here may be valuable complex eigenvectors.
Just a sample show,
Consider the following matrix; i minus the square root of a virtual number to be.
This means you get the equivalent matrix where we see good
I will put a minus, minus happening here, this is also a plus i is happening.
You see, when we get it overturned a plus i will go up,
A minus sign will come down and will obtain the same matrix,
that is to say a matrix type of a hermit in the matrix.
Our claims if there are any how complex value
eigenvalues of this matrix will be real.
Let's do it.
Likewise, we are writing the matrix
and we put on the diagonal minus lambda.
Now let's make it the product, see the product lambda squared
lambdal terms; there is less oxygen here,
plus lambda are here, lambdal terms fell, remained zero.
If we look at other numbers you have see a minus here, with a multiplication,
plus i multiplied by a minus if one,
A negative comes with a minus sign, because i think i hit minus one off
It means that there is a minus but a plus i'l is a minus to a plus ii
hit the mark two turns, but also because we get minus on the second diagonal,
means that this matrix will be the determinant of lambda squared minus three,
a minus here, minus two from here in terms lambdal
dropped frames, but minus lambda lambda lambda plus times
where he falls for the first lambda force, the force functions of this happening.
That we are equal to zero when it turns out one of two root root
three one minus root three.
As you can see, even if their matrix elements of the complex
but there is a special structure complexity, takes the complex conjugate
When you calculate the equivalent ousted him coming, it's real eigenvalues.
But we see that the real coefficients because of its eigenvalues,
coefficient in the matrix, the matrix is complex, we are writing again,
bring three roots we put on the diagonal,
we arrive at an equation in this way.
Eigenvalues again that it opposed,
XY components work here say you get two equations with two unknowns.
Yet these two unknowns, as always, there are two equations here are some more
Although the visual equivalent of each other hard,
UC hit the root of a positive first equation, the second
If you hit a plus i'yl the equation, you can see that the same two.
Now here it means that there is still one individual against two unknown equation.
This is independent from the equation X plus i know you say easily,
plus i here you can easily find what Y.
You can find three minus root as a
because when you say there is one X plus i plus i.
Then three times a minus X plus Y is equal to zero,
According to the X means Y is certain, plus i had fallen
According to a minus root three of opposite sign, so a stem has three minus.
When you put two in the second to similar roots
Or, you find you do it in the e bir'l
You can also find easily to the two's relationship
Assume that this delicacy, but we enter here, we find two of e.
that is equal to the two lambda lambda for two to three minus root also,
this is a real value, it's time we put on the diagonal
components of the vector X Y can still say,
so these are not the same as above, X-Y,
This temporary XY values are chosen to do just accounts.
Two equations are again, again the usual case,
these two equations are not independent from each other.
One of them,
So voracious are able to solve the unknowns
for example where X is still a choice we do
wherein Y
choice seems easier to do that than here because we want to get good,
I say a minus Y, where X is the
three minus root as you find one, you can check the account details.
Now what are the features?
We call this vector are perpendicular to each other,
valued vector in the complex conjugate of one of the first product, the complex conjugate
When we think of as a shock to the voracious domestic product.
to a complex conjugate,
You also will make a line perpendicular to this
If you hit the account with these two vectors will see that zero out.
So here in the vector is able to bring to each other.
This can bring the unit vectors of length,
You can also details like that.
So we finish up this lesson,
I encourage you to make these examples alone.
These square matrices, find the eigenvalues and eigenvectors of the square matrix.
Numbers are selected simple.
There are many zero as you can see, there are triangular matrices.
The basic concepts of encountering difficulty with this account
In order to test and accounts habits
I suggest you do to gain matrices.
As you can see, this matrix is not symmetric.
Then you can encounter different situations.
That may not be the real roots, repeated the
When eigenvalues sufficiently,
eigenvector found here may not have these features.
In the second set of questions
The given symmetric matrices, these symmetric matrix
opposing [BOŞ_SES]
We want you to find eigenvalues and eigenvectors.
As a summary in each section
the eigenvalues and eigenvectors position here.
There are three main themes in fact, the concept of eigenvalues.
Eigenvector following; that when you get the transformation of a vector
generating the transformation matrix coefficients.
But style is not such a collection of eigenvalues
just to give you a time to one it turns out,
When you give to two e two turns only size may vary.
And who here only numbers on the diagonal
but of course we can not know in advance that consists of a matrix.
We are looking at how to calculate it.
Now the concept of the matrix, wherein the first means
the square is not gerektiy
A fun because you can have against each e.
When you make the transformation to a size other than me this eigenvector
the concept of eigenvalues and you can not, but it square in the matrix
It can also base it on that target have space on the right
as in the definition of a vector space it is possible with the same selected.
That's equations is calculated in two steps.
Eigenvalues has before, the determinants of it negative lambdal
We find we find eigenvectors and eigenvalues in watching it.
So this is actually the second theme,
What could be in symmetric matrix in the third theme.
Without roots can be as complicated as symmetry can get the real roots.
As can be split roots, but the roots can be repeated symmetrically
In non-matrix, when repeated roots
There is no guarantee of finding adequate eigenvector.
In the interest only eigenvalues real symmetric matrix, complex predicament.
Repeated eigenvalues and eigenvectors, even if they are opposed to,
a sufficient number of available eigenvector.
I encourage you to pass this on in the abstract.
Another important feature of a symmetric matrix
together vertically to the eigenvalues.
Hermite matrices work somewhat generalized symmetric matrices that,
complex coefficients may matrices.
For now, goodbye, now in this cluster,
In the course set, the subject matter in this set
We will see this practice after seeing the problems,
The first of these applications comes to the diagonalized matrix.
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