Simetrik Matrislerde Özdeğer Problemi / Eigenvalue Problem of Symmetric 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

13 ratings

Koç University

Doğrusal Cebir II: Kare Matrisler, Hesaplama Yöntemleri ve Uygulamalar / Linear Algebra II: Square Matrices, Calculation Methods and Applications

13 ratings

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.

From the lesson

Kare Matrislerde Özdeğer Sorunu / Eigenvalue Problem in Square Matrices

Öz Vektör ve Özdeğer Kavramı / Concepts of Eigenvector and Eigenvalue13:28

Özdeğer ve Öz Vektörlerin Hesaplanması / Evaluation of Eigenvalues and Eigenvectors9:22

Çözümlü Problemler / Solved Problems34:35

Simetrik Matrislerde Özdeğer Problemi / Eigenvalue Problem of Symmetric Matrices9:22

Hermite Matrislerinde Öz Değer ve Öz Vektörler / Eigenvalues and Eigenvectors of Hermitian Matrices12:42

Meet the Instructors

Attila Aşkar
Prof. Dr.
Matematik Bölümü (Department of Mathematics)

[BOŞ_SES] Hello.

We solved five examples in the previous session.

The purpose of the selection of these examples was to demonstrate conditions that may arise.

These were examples of simple numerical terms.

First and fifth in two binary matrices,

the other three were also three three-pointers matrix.

In the first example, we found two real roots.

Ie two in a binary matrix.

Again, the two binary complex eigenvalues in a matrix of two valuable

We have seen can be found.

In the second example for each of the three eigenvalues out and we found an eigenvector.

The third and fourth examples of a second discrete eigenvalues found

eigenvalues was a repeated root, but it was a different situation.

In the third example, although one of the two eigenvalues

Repeat if we can find time to opposing the two eigenvalues the eigenvalues.

Therefore, the non-recurring one root

In contrast, we find that the eigenvalues full three independent eigenvectors.

So our ability to build a base emerged.

But in the fourth example of a repeated eigenvalues

We have to find a single one eigenvector.

Therefore, we find only two eigenvectors.

Therefore, a basis vector for a three-dimensional space

We did not have the ability to create a team.

Now there may be a problem with the eigenvalues of leaving the complex.

Because in many cases the eigenvalues of a physical magnitude,

It may indicate a probability value.

For example, a power, a frequency, a probability

With numbers like real value, it is as valuable real definition.

Therefore, to ensure that they are real valued matrix,

We saw it in the matrix in which they must be symmetrical.

So what should be in them as a guarantee

We will ask you out real valued.

Again sufficiently eigenvector

What is the find of the requirements he will ask for it.

Their answer lies in symmetric matrix.

Now, at the beginning we do not know yet, but we will prove the following:

Real roots in the matrix will be symmetrical.

Even if repeated roots, they sufficiently

We'll show would eigenvector

and the eigenvalues of this would be perpendicular to each other, we make happen.

So, there are some supremely superior properties of symmetric matrix.

Now, before you go to the public our önsezgi

In order to strengthen a small matrix start, half a binary matrix.

Here you have no condition on the coefficients.

a 1 1, a 1 2 a 2 1 ,a 2 2.

We are writing to them, this matrix.

We are removing the negative lamdba across the diagonal.

This will give us a determinant eigenvalues.

Indeed, we have here see a single lambda squared term.

Multiply the two numbers on the diagonal,

lambdal the only minus lambda times a 1 1 here,

minus lambda times a second, including two from here, lambdal have these two terms.

And lambda are not available, the absolute numbers

a 1 1 a 2 2 eksi a 1 2 a 2 1 terimi var.

Now we find here the lambda.

These are the roots of what happens when we ask the real answer immediately clear.

This quadratic function, power function

To find the roots of this work A1, B number here,

c squared minus four times, including in a number wherein a,

c plus has to be valuable because it will take the square root.

We are writing here to be square, a 1 1

A 2 minus 2 squared term here four times.

We are witnessing just this: a plus from a 2 1 1 2 1 1 frame work,

2 2 1 1 2 2 squared plus twice that.

But here comes the same term minus four times.

So 4 minus 2 minus 1 will be a twice-to-1 2 2.

Birde a 1 1 kare var a 2 2 kare var.

So when we received the following three terms a 2-to-1 1 minus 2 squared off.

Back at plus 4 times a 1 2 2 1 off.

Now we just ask this question: here we find the roots of the subject,

1 plus 1 plus the square root of minus 2 2 n the D D

We find their roots in the root.

But to be real roots, plus the D's got to be worthwhile.

We have a complex value to be less valuable if D under root.

This D plus the only way to ensure that valuable.

To guarantee in all circumstances

a means is equal to a 1 2 2 1.

Because in this case the first term will be a sum of squared plus another frame,

This will give you an absolute plus valuable.

But this is not necessarily required.

Because a 1 2 2 1 plus can be valuable.

Here, too, there is added value.

So a 1 2 2 1 may be equal to the value without the plus or

there is a surplus.

This product can be the product of two very large negative value.

He took time minus the D.

But in any case, a sufficient condition is equal to a 2 1

If you can not guarantee this in a 1 2 D's plus is worthwhile.

a 2 1 from 1 to 2 means that even here we go to the matrix, they

The diagonal matrix because they are in the equivalent position is meant to be symmetrical.

So here even just half of a binary matrix symmetric matrix

We show that the roots of the property plus enough to be worthwhile.

This does not mean that the roots will be absolutely symmetrical least complex.

Roots again, plus provides, may be real valued.

But if you can not guarantee symmetrical.

But the real roots of symmetric can not guarantee it would be valuable at any time.

We did not prove anything more here in half a duality but also our intuition

developing symmetry means that there will be something she could imagine.

In fact, the next two theorems,

n times the size of the matrix in this

It will indicate validity of this observation.

Theorem as follows: a theorem saying that the real eigenvalues.

A real, symmetric matrix of a matrix that is symmetric and

If the eigenvalues are real valuable real.

Eigenvectors for the eigenvalues be real is real.

Now how do we prove it?

Go with a proven this in reverse.

Please Suppose we say that the roots are not valuable, but the real symmetric matrix.

If a real root valuable time will be the alpha plus beta ii.

Other root must be minus the alpha beta.

Because here we have the account of a negative determinant of lambda i.

A force that numbered function of real coefficients for everything real.

This is a root complex and the other one is surely to be complicated.

We have seen them in such an example in the previous session.

Now that these eigenvalues lambda lambda and a star.

Let us write that equation.

When we found here will be eigen to the lambdayl.

When we found to be the stars to the stars.

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