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 Matrislerin Her Koşulda Köşegenleştirilebilmesi / Diagonalization 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
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From the lesson
Matrislerin Köşegenleştirilmesi / Diagonalization of Matrices
Meet the Instructors
Attila AşkarProf. Dr.
Matematik Bölümü (Department of Mathematics)
[BOŞ_SES] in the previous session,
We have seen that the general diagonalized matrix.
And we can find sufficient eigenvectors of the matrix,
as the one we are building the Q column using these eigenvectors.
He was taking the inverse of the Q diagonal matrix obtained from this product.
We found this in the overall matrix.
In analyzing the eigenvalue problem, we have seen a strong side of the symmetric matrix.
He is strong sides as follows: a symmetric matrix always
We could find enough eigenvectors.
This eigenvectors
column as always so we have a chance to use
and therefore can not always symmetric matrix diagonalization.
[BOŞ_SES] the results we obtained from this general theory.
There are any more of a convenience alone in symmetric matrix diagonalization.
That is: We are situated, we find Q as columns the eigenvectors.
But if we choose to be one of the necks of these eigenvalues,
You did find a vector because it can always
e1 length of time the unit vector 1 if you divide the neck would be calculated.
Q. If we do it this way, we get to see Q transpose
When we get a matrix with rows to have.
How would this matrix, now that we determine what the property.
Because the columns in a symmetric matrix to each other,
We always knew that even if other vectors repeated eigenvalues.
We did Height of the unit.
So different display to each other and perpendicular to each other in the column.
Inner product gives zero.
We get the first time we take the inner product itself.
This, multiply the transpose of the Q q we use this feature.
See, that's where the line e,
matrix column e, which means we stood matrix.
This product follows: e1 brought to the horizontal position we're taking hits to the first line.
There is e1'l e1 of the product, that one.
We bumped into a second-line, third line we stood, we stood n'yinc of the line.
All other zero because e1 others.
This means that there are over one diagonal in the first column, there is zero elsewhere.
Similarly takes any column
When we hit that turning the other but multiplied by their
This means that only time will give 1 consists of 1 on the diagonal.
Its outside is reset occurs.
So we know that the Q Q transpozl unit that matrix multiplication.
On the other hand matrix that each unit gets hit
two matrices are inverses of each other.
Here we see that just Q transpose Q is equal to minus 1.
We call this feature the unit orthogonal matrix to the matrix.
English Orthogonal also called on the Turkish
Pardon is called orthogonal orthonormal, orthonormal also called in Turkish.
Ortho're already planting.
Sorry to say the unit.
Normally, it normalized to say, the matrix.
Here now the following theorem into account, we demonstrate.
If this Q column units of paint made from them
the resulting matrices, the matrix Q is an orthogonal matrix unit.
Transpose because he, unlike equals.
We also calculate the following: general rebuff to the Q
He was getting hit Q'yl to hit the diagonal matrix.
Symmetric matrices because it is equal to the inverse transpose of the Q
making it easier for multiplication.
Why?
Q minus 1 to account quite often want a job.
But when we create the columns of Q unit vector
Get it devriig by rows of columns did,
So we see that the inverse transpose of Q found.
Therefore, we can find that almost inverse matrix without any processing.
Such a feature of the symmetric matrix and always enough
degree to which we find columns can do it all the time.
Q Here in the eigenvalues
It occurs, but we have made unit length vectors, eigenvectors.
This has obvious if we look at the visual matrix of the matrix structure but symmetrical.
a 1 2 a 2 1'e eşit, a 1 n a n 1'e eşit.
Symmetrical like this.
We create the right eigenvectors as columns using Q unit.
We use the unit eigenvectors as still left in the line.
This challenge multiplied by this matrix, the matrix is symmetric
We come to see that the diagonal structure and the eigenvalues on the diagonal.
Now we want to examples.
The first of these examples,
A simple two binary matrix.
We need to find eigenvalues and eigenvectors.
We are writing on a diagonal matrix minus minus lambda lambda.
We solved this problem already before similar problems.
194.
There are problems in this page.
We find it easily eigenvalues.
Lambda squared minus three lamps,
less than three lambda lambda minus six and minus nine
We find an eight functions such as power,
Eigenfunction the eigenvalues of that function.
This value of a second order is two and four
We find the roots for the formula we know that the force function.
Here we give the roots.
We provide this easily.
When we received two of the four, minus six,
six times minus two've had 12,
There are four here say here that there are eight plus eight plus four,
plus 12, minus 12, gave zero.
We saw that provide four.
This includes opposing these eigenvalues eigenvectors given case here.
Now there are two ways.
To diagonalization of this matrix.
This e do not touch any of them to see not a boy.
e1 square of one's height squared plus one of the two, the square root of two.
So long root of two.
E2 also two long roots.
We say that in a way the general theory
by moving the first column e1 second column
We calculate the inverse of the matrix formed from the E Type E2.
You can really see that this is the time we hit the unit matrix,
This is calculated correctly.
Q 1 and Q minus to the middle put the trio
product warranty will be taken by diagonal theorem gives us.
Üşenmeyip ensure we do this we find the multiplication.
But no need to make this product because diagonal theorem guarantees us
will be on the eigenvalues, the reset will happen and outside köşeeg
e1, e2 to be here in the eigenvalues in opposing,
e1 diagonal for the first two would be the number we received in the first column,
theorem guarantees us four would be the latter.
We can do that in a second.
e1, e2 vector by dividing the length behold a
a negative aspect because it was splitting into two long roots that we find Q.
When we get Q transpose it is no longer the opposite.
See if you multiply the first line of the first column of this product.
One half plus one half, take one.
If you hit the second line plus minus zero.
Cons merger, given the supply see here,
We see that when we made this product the unit matrix.
So this theorem sums us.
Already it to deliver the product,
You do not need to do but this is a useful exercise in terms of sheer see.
Again this form of Q transposed,
Q. We know diagonal matrix that is obtained by multiplying.
Q minus one important win here to find no need to take action.
Q When you get the tactics right inverse matrix transpose.
This of course brings a significant gain account.
Dualism, but of course we can do two opposite matrix grows
get in would be a great process in itself.
However, if you create from a neck Q vectors
We have found just the opposite by taking the transpose Q.
Now, let's start again a symmetric matrix.
We solved this symmetric matrix before,
We solved the sixty-nine hundred and sixty-six hundred pages.
There are two repeated root, there is one discrete root.
These vectors from where we choose self-contradictory.
We have seen that it is easier to work bringing the unit size.
Meant to unit size, take these vectors to divide their size.
Here they were brought to the unit size vectors.
Q. We'll get them.
Using these vectors as columns in size
and multiplying the
Q is the matrix transpose that he slammed unit
By providing direct product we can find.
Six still on the diagonal by the triple product,
We can direct the future of six and twelve points.
Just because we do not provide, we also let in concrete,
You can spend some time here this triple product, but the power of the theorem,
It guarantees that you will come to this account without the diagonal structure.
Also possible in the Hermit matrix diagonalization.
You will remember the outside diagonal matrix Hermit
which elements would be valuable complicated.
The square root of minus one that im here
The complex conjugate of the matrix
If you receive income equal ousted him away.
We've seen this before with one hundred seventy hundred and seventy-third page.
Even though precious complex matrix,
Core values that provide the structure for the Hermit
real tactics.
But this is not the real essence of the vectors from opposite to the core values.
If we divide the size of these transactions is a bit more complicated by
then with this crash, see, the only difference here just transpose the
not because the first matrix multiplication in a complex valued vector
or the need to take the complex conjugate vector.
Multiplying this challenge, again, we see that the eigenvalues on the diagonal.
Now we see that quite a few calculations, we have seen examples, we saw theory.
Now, let's take a break.
After that we will continue with the transformation matrix.
Goodbye until later.
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