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.
Matris Dönüşümleri / Matrix Transformations
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Doğrusal Cebir II: Kare Matrisler, Hesaplama Yöntemleri ve Uygulamalar / Linear Algebra II: Square Matrices, Calculation Methods and Applications
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Matrislerin Köşegenleştirilmesi / Diagonalization of Matrices
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Attila AşkarProf. Dr.
Matematik Bölümü (Department of Mathematics)
[BOŞ_SES] Hello.
Our preceding a session
Q inverse matrix to the left,
We see that we are right when we hit Q'yl obtain a diagonal matrix.
Here, we are creating the eigenvalues of matrix Q.
Similarly, the matrix is symmetric,
Q Q. We saw just the opposite can be obtained from the overturned.
Here Q columns of the matrix modified in length units,
organized unit consisted of eigenvalues.
Now it is actually a transformation process that we do here.
We transform this matrix into a new matrix with this kind of product.
What we saw at the end of this transformation of our previous one
diagonal matrix was to occur.
But if we do not create the eigenvalues of matrix Q,
This will not necessarily transform a diagonal matrix.
Similarly, a symmetrical matrix.
Therefore, we consider overall product of this kind,
triple products, the matrix obtained by multiplying the left and right
A new matrix is, it is the result of a transformation.
There are two matrix why she could imagine.
Why did he have a right matrix The two left.
Because the definition of a target matrix space
It serves as a work space in a vehicle.
Thus one definition of jobs in the space of the matrix on it.
One of them is also in the target space.
We need two matrix.
Considered from another perspective,
We say we look at the matrix elements ij that there are two indexes.
Right affecting any one of the index on the left.
Therefore we need two matrix.
So if we just AQ Q Or, if we minus 1 Or,
Q transposed into a transformation if we do not provide this.
Both will be right and the left.
To sat on it a little more, rather than to reconcile a suitable structure
Consider the following equations for the n unknowns.
x unknown, the coefficient matrix,
Right known as B.
Now we if we return with a Q matrix x.
x'e Q kere y desek.
Vectors sees a matrix job.
Because the vectors are located in a single space,
If we can convert it acts in a single space for him.
In case it is not because the matrix of matrices and a definition
We need a target space and space.
There are two matrix.
A matrix of vector a Q enough.
B to Q must not necessarily be the same so that a P us transform.
This indicates that B is derived from a c.
Let's put them in their place.
We say x instead Qy.
See, instead of a time we say x equals B times C P.
Now we assume that there is the opposite of P in this transformation.
If we hit the car on the right side just opposite the P
but remains left in a P minus 1 stays revenues.
A more general transformation of this transformation as you can see above.
Because the vector x, the definition of the space we convert the vector x Q,
We also convert the space in the target vector b P.
Now, of course, such as a PR
P P and Q not work so well for his practice
As you can see we get instead is happening Q Q minus 1 times.
This type of transformation we have seen before.
So we would remove it.
We have a better structure's put.
In fact, we hope to change,
not convert the matrix to start with.
A'nin etkidiği x,
it is the definition of space, it brings to b in the target space.
So we x the targets, the definition of space,
We are going to b the target space has turned into converting.
First starting point from which we naturally can not convert to,
It is undergoing a transformation itself.
We give them names.
If the left to right in a Q reverse a
Q If you get hit with a new matrix.
This transformation also called similarity transformation.
For a second embodiment the secondary
Let's force functions.
As we are creating.
X. Right-to-left line in the x transpose the matrix,
When we hit the right column matrix of x
quadratic function in terms of components, we were getting.
As visual structure of it.
Standing in the middle matrix.
Or, right column matrix column vector x,
on the left we see that toppled be tilted so that the t
again, this time as a vector x line matrix's rows Or, vector.
This product in terms of its open software components
when we made the aija x and xy are multiplied.
As you can see quadratic term occurs.
These functions.
Now if we return we xi,
Or, if we so a return from x to y y
We can think of it as an x, so if we settle,
x instead Qy; Let us on this side because it has the opposite xt.
If we get it overturned, you know multiplication
It remains as it comes, but changing.
X transpose this year as well as to transpose Q transpose
Let's place for it.
This is f (x) will be a function y in terms of function, it instead.
Qt the yt xt we put it here instead.
It is standing in the middle.
Sagya also, we put the X on the right side Qy.
If we change the order of these operations,
Take out the variables.
Matt, let's reveal the triple product of the matrix.
A Q Q transpose the right and as you can see here
It arises in a natural way by multiplying the left.
Here we get a new matrix.
If you are here on we symmetrical symmetric matrix.
And reduced size of the unit to the eigenvalues of Q,
It was brought to the arranged eigenvalues
If this transformation will give us create a diagonal matrix.
But if this works any Q
We will call this matrix and the B matrix.
We call this kind of transformation the equivalent conversion.
Because we translate the Q wife.
Here we emphasize that called the transformation of congruence.
Now we can ask the question.
Q with the right, the left, we hit the Q and vice versa.
We have achieved a B matrix.
We hit the same to the left with the same Q and Q transpose.
This time will generally be of course a different matrix.
Now I wonder meaningful question, because a little more
We know from previous studies that a positive response would be,
I wonder if B and B will do the same and two of the base
Q Can we also can choose to diagonalization.Lines might ask such a question.
So we can not deal here with two different transformation with a single transformation,
we can handle our issues might ask.
The answer here is clear.
That's what we get if we Q minus 1 Qt.
This brings a new kind of transformation that is more effective.
if Q is equal to minus Qt's called other conversion units,
and the transformation that it provides
the other unit conversions, allowing it to matrix
We call the other unit matrices, so there is a small detail.
Q We call him the other unit matrix, the transformation they provide,
QT Q minus the usual situation where an equal,
other unit conversions, or also called orthogonal transformation in Turkish.
Just a theorem, the determinant of a conversion unit upright
or a minus and vice versa is also true, the opposite of this theorem,
determinant of this matrix is plus or minus one unit is orthogonal matrix.
Just to prove to me, a very simple proof.
Now that Qt minus one equals Q Q Q QT times times will be a minus.
But Qin Qin times the identity matrix minus a definition,
Q transpose Q with the product means that the unit will be.
We have seen this theory on the other hand; if
If a product of a matrix,
With B, and if we account of the determinants of this product
B would be multiplied by the determinant of the determinant.
Q. Now, once we apply it here
Q transpose of the matrix determinant of unit
will be determinant, since an equivalent product.
But the Q transpose Q determinants equal
According to one because of the expansion of the determinant column
It gives the expansion of determinants by rows One,
Q line and because transpose mean change column
If you think that you can invest it like a square block matrix
Lines column, the column is going on line, changing this.
Q say that the determinant here
a squared is equal, when we solve here the Q determinant plus
As a proven theorem or minus, and vice versa.
If you go this road opposite end of Q'yl Q's ousted
We find that the product should be the identity matrix,
This also proves that the orthogonal matrix that unit.
But now those with a minus is a plus determinants
This should be a difference between the difference in the definition of these two states; any one
called a rotation matrix determinant equals, we will see an example of it right away,
coming to a minus determinant is equal to a reflection matrix.
The shape takes a shape in the first
comes opposite to be returned intact,
one in a little more detail in the image
It shows that it opposed.
Now let us prove a theorem for them.
The unit protects other product and vector conversion inside the neck.
So if we return a vector x u,
Or, if we reverse the rotation of the x,
Let x height into account.
Q multiplied by the length of the QT interval x.
Now that x = Q xt;
UT QT would u x is Q times.
But we have other orthonormal matrix,
other unit matrix, ie, a product of QT Q,
Or, from another perspective QT Q minus one equals.
So where is the data matrix multiplication unit, we're writing unit matrix.
Multiplied by the unit multiplied by the matrix means that ut again.
According to give the product of the unit matrix M ut ut once would have.
So XT times x is preserved as ut times.
xt ut times x squared times the length of the length of the frame,
means that did not change in size has remained the same.
As we look back on a bird's eye view, between the X.
but this transformation other qualified conversion unit conversion.
We see that at the end of this transformation x length change,
because the length of the new vector x of length are the same.
Similarly, if we take the inner product,
i.e., x is a u
Q'yl we're into, we've also convert y from a van.
Taking the first of these two product inner product
We think as a line, in the second column, does this inner product.
instead we place the XT X. ut comes QT.
y here because we need to y, Q times have come.
Once again the same mind QTR Q
Where Q for other conversion units,
because the matrix gives the identity matrix.
E v is the unit matrix multiplication, the product of any vector still gives v,
because an objective matrix matrix unit,
If you hit what does not make an impact, a neutral matrix.
Here comes the v multiplied by ut As you can see,
u'yl in which the inner product of v.
This is why it's important, because it did not change the length, the height v Similar
the change, means that the inner product of u v u'yl in length,
v The length multiplied by the cosine of the angle in between, we know the inner product.
likewise X. y,
x y height times height times the cosine of the angle in between.
E whereby x height remained as the length,
v y long as I stayed in size, so do not change the angle.
This means that only one other intermediate conversion unchanged shape as the object
It is a way of transformation of the size and terms remain unchanged.
The same thing can be shown Hermit valuable for complex matrices.
I do not want to enter into this same thing is done with the same name,
Hermit matrix in a single complex value difference,
namely the definition of a complex symmetry
We have to take the inner product of the conjugate and
this complicated, complex conjugate in the future.
All of these cause important because if we say,
a basic knowledge of the application,
It remains invariant under rotation and translational laws of nature.
You axes as you take your hand I drew my team,
Even if you do a physical problem calculates the coordinates
It is necessary to remain the same even if you change the result of the account team, or my
Enjoy lo different results according to different preferences that nature would rule.
In Galilee it called invariance.
English is also Galilean invariance he said,
This is a very important natural laws.
Let's do an example.
Let there be a vector x,
I turned around as an alpha of this vector x without changing the length.
This matrix allows him to work.
You can find with this geometry, you may have seen in your other classes,
You receive the x component of the vector x, y component, the second component
You receive, as of yet, you receive the first and second components of the vector y.
These are respectively a y x a x y two and if the two
the x coordinate is rotated until the alpha but long
This is a change between the coordinates of the vector y unchanged.
As you can see here is the determinant of the matrix cosine squared minus,
sine-squared equals minus one plus more for the next one.
This is a rotation matrix.
No need to think about it too, because the definition of the determinant bir's
We have seen that change shapes and angles.
But here let's do some more detailed geometric calculations.
Here we do an x-y multiplied by the cosine alpha once
sine alpha minus two times, and two years is achieved with a similar hit.
Now that we have the y component of the vector y
Let the vector length calculation results obtained in this transformation.
Of course length,
frame of the first component and the second component; Pythagorean theorem.
We saw Y 1.
The term in the following brackets.
We take this to the square.
Y 2'yi de gördük.
This term, which is fully in the end of matrix multiplication
evaluating the results we found, we find using.
Now let's open them.
Of course we open the first term squared plus a second term
Multiplying the term because it is squared minus times minus two.
In the second brackets together in the same way but it has a plus sign.
First it squared plus second squared multiplied 2 times longer.
Notice that stood where x 1 cosine, sine 1, where x stands.
Here was çapıy sinus x 2, wherein the cosine hits.
When we opened them, we see a very interesting result.
See, here cosine squared x 1 square stands where sine square.
We will collect them.
This means that the output x 1 squared here.
Where x 2 square of the sine square stands where the cosine square.
When we gather in the square x 2 output.
Again, it was a pleasant thing.
Mixed product where both x 1, x 2 and sine, cosine multiplication
It comes with a time axis minus sign because there comes a time in the art.
They are taking each other during the gathering.
As you can see, only the sum of the square by square x 2 x 1 remains.
Y on the left side of the square there.
This more compact, as we wrote a collective statement,
y the length of the frame, we see that x is equal to the square neck.
So under this transformation,
When we turn, we see that y x length x length remains the same.
This is already proving we did before, in general,
proving the theory is applied to vector in a private plane geometric form.
But I hope he gives a more concrete understanding.
Similarly, let the projection matrix.
See, wherein a bit different.
The other had one of the cosine and sine-cosine axes plus one was.
Taking this product, we calculate the determinant of this matrix,
As you can see, minus cosine squared.
It comes from the product but the plus sign will değişter.
Determinant because we see that the sine-squared minus -1.
So, this other unit conversions,
other units of determinants of matrices but this is happening for a reflection came to -1.
It comes opposed to this; x returns before you as an alpha angle,
as in the previous case, after which the vector x
You receive the reflection relative to the axis.
If we make this product, namely x from
Next we go to, it's reached here.
After a rotation in a reflection geometry from that thereof wherein
I do not want to tell him for a long time, spend time, but I do not want the case.
Now them again, y if we account neck,
here the length of the first component squared plus Y squared second component.
Recently quantities obtained through the conversion.
When we opened this square is happening again as the previous one.
See, once kosinüsl to x 1, it comes at a time sinus.
X 2 as well as similar.
Meanwhile there is less in the first term.
In the second term, because we can say that there is plus minus minus
When we take the square will turn into a plus for both.
Again, the cosine squared x 1 of the square and the square sine, square coming x 1.
Similarly, in the coming x 2 x 2 square of the sum of the squares instantly.
This summer we even collectively,
y is the length of the left frame,
here we found a frame of length x.
They are of a general nature.
We have done here in two dimensions, but any
condition Q determinant -1 or previous situation
As we have seen is that both conversion +1
lengths and angles,
We proved earlier that protects the angle between the two vectors.
To see where the theorem,
We prove this theorem with the 207 pages in general.
We're not talking in a way that no previous proof of our size.
Judging the relationship between the rotation and reflection matrices, see,
we examine here the first matrix, rotation matrix.
As an alpha returns.
After you return to this alpha as you keep the same first component,
You receive the negative of the second component.
But, see, where do you keep the same first component,
You keep the same return after the first component,
You receive the second component of the opposite sign.
This unit matrix is not at one of the corners +1,
-1 matrix is provided with one.
So, these two matrices together and watching it spin
We see that as a mirror attached.
Now, I thought this was an interesting topic.
After that will come a subject like invariants of matrices.
So we're doing the transformation of this matrix, whether this transformation
Is there a change in the information during inherent in this matrix?
In particular, we can ask the question: are such a similarity
What happened when you make the transformation diagonals?
We will see this in more detail in our next session.
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