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Merhabalar.b in the previous session briefly tell you about what they want to do the course I

and I gave a summary of the issues seen in a section.

I hope you can see them.

Perhaps not immediate.

In order to understand these issues because part one, a prerequisite, not a prerequisite.

But they know, the matrix below them

it is important to understand the underlying.

Now here we are two parts of the determinants and inverse

matrix eigenvalues, eigenvectors, diagonalization, matrix functions

and we will discuss the application of matrix functions.

Main lesson we here we go.

Now we'll see square matrix in chapter nine.

Why is there a square matrix and explain the benefits.

Because we have the privilege of a square matrix.

To understand this we have to see what happened, some operations.

After that he came from the concept of determinants.

But such a determinant determinant as possible to give definition

It has emerged as fantasy.

A concept that emerged spontaneously in response to a basic question.

Where it came from, and even the meaning of words,

determinant in Western languages to say it came to determin the subject,

We call also on the deterministlik philosophy.

It means to determine the determin.

So we have a decisive role.

Determine what you'll see.

Before these two equations with two unknowns and three equations with three unknowns

We observed that after seeing the determinant naturally arise.

Some of these determinants of observation to find the basic rules,

called a Laplace calculation chain to the calculation method

We reached here after determining characteristics and determinants of the well

We will proceed to identify and route calculation.

Let's start with the definition of a time frame matrix.

We call matrix of rows and columns to equal the square matrix.

Now two

We have seen in other places, but you know a part two

means to collect matrix means to collect the number of opposing each other but

because it is not always possible to make every

A number of the matrix has to be the number one opposing matrix B.

This line of the row of the other one of these two matrix

and it must be equal to the number of the column of the B column.

You can collect the same matrix rectangular matrix but hit

If ye would çarpamazs this time because you have to crash compatibility in this time.

Therefore, we say let both hitting the same in both collection,

B matrix and both have two rows of square matrices and

and we see that the column should be equal.

Hence, the possibility to make the process a lot more in the square matrix.

Let's say you bought two rows and three columns, two two matrix,

You can collect.

But if you go up there hitting your çarpamazs.

That's why the square matrix provides more application possibilities.

[empty sound] and possible in the square matrix

but without the ability to define some features in non-square matrix.

We can not define the square matrix symmetry and anti-symmetry.

We will describe them quickly.

We can not identify non-square matrix in the matrix unit.

Find the matrix which is the inverse of each other

and it would not be able to find one as non-square matrices.

We can not identify determinants.

We can not define self-worth and self-vectors.

We can define the diagonal and triangular matrices

and multivariate quadratic function in such variables eighty

quadratic function as well as we do that in many places

uygulamalarda.b may need the easiest and most efficient,

We can do effectively square matrix.

The function of a matrix.

For example the frame of a matrix, a matrix of a matrix nth

but we can define the functions such as square matrix tops.

In this respect, it is providing the opportunity for many transactions square matrices in many areas,

A wide range emerges as a powerful tool.

The most important application areas equations.

Algebraic equations even with the number of unknowns in the equation equation

An unknown number of cases is not equal equation

Even the square matrix can be reduced.

Core values and core functions, core issues such as vectors still frame

taking the matrix and matrix functions.

Now let's get more concrete structural issues that we have said.

When we take the transpose of a matrix that is equal to, if he comes

Transpose that the T icon also comes from T'Su transpose the words,

We call it symmetric matrix are equal.

This reveals that the; A matrix

elements of transpose, to the elements if they aija

You change because it is ajia transpose rows and columns mean.

Even when we change column with this line

come on by diagonal matrix

According to the equivalent row and diagonal

the number of positions that reflect the diagonal equal say in the location.

For example; Let's go over here aija.

i have a first line because the first line of the index,

The second column indicates the index.

A first index i.

When we come to the second element of the first row number two so [1 2] element.

According to the two symmetrical positions

show that the index in the second row a good two QJ.

So [1 2], the [2 1] comes equal.

This is where visually similar to the minus three

[1 3] equal to that in

A [3 1] The numbers in .Bur equal.

It follows the visual understanding.

You draw a diagonal,

According to the numbers should be the same in each symmetrical diagonal position.

According to these two symmetrical diagonal in this position.

These negative numbers came in a symmetrical position with respect to three of the diagonal again, minus three.

We can not say anything on the diagonal, because those in diagonal stays.

According to the zero you find reflecting diagonal

-equivalent position in that number is zero.

That means that the matrix and the two

When you change the column with hand lines

The inverse transpose of the matrix that you see is equal.

Anti-symmetric matrices in the opposite situation.

Transpose the matrix itself minus

This also means equal to transpose the marked

We will modify the elements in the row.

First and second place değiştirceğiz from aija indices.

It comes to the aij'n the minus sign.

When i and j is equal to the right once on the diagonal

See figures wherein [1 1], [2 2], [3 3] numbers.

I collect them last time-out is equal to minus ai.

This shows AIII is zero.

So the numbers on the diagonal are zero.

Again this is based on the reflected diagonal

The numbers in the position we find that the opposite sign to each other.

See, according to these two symmetric reflecting the diagonal

When you find the position that location here.

Number here minus two.

Similarly, according to the three symmetrical positions where the diagonal.

Here we see that to minus three.

Zr, here again, the four points where the four numbers

According to the equivalent position of each diagonal.

But one comes to be missing one of the cross hair.

This brings the antisymmetric matrix.

We will see that a team of special forces symmetric matrix.

Diagonal matrix as follows: The numbers on the diagonal are non-zero.

But out on the diagonal, all the numbers in locations other than zero.

Some may be zero but all of those over the diagonal becomes zero

anyway, the whole place comes to your face zero matrix is zero.

Therefore, a feature of the matrix

and we see that it's got to be a matrix, which is only possible in the square.

Because if square matrix is a rectangular matrix to see if the right side,

There is a diagonal rectangular matrix.

So even if your gonna draw diagonal geometric,

then the numbers start to come on the diagonal.

Diagonal matrix slightly weaker

feature matrix in terms of symmetry, triangular matrix we say.

This is useful in a matrix type.

The numbers below the diagonal is always zero.

The numbers above the diagonal are non-zero.

Some may be zero, but it's all happening already diagonal matrix becomes zero.

It would be opposite.

Numbers below the diagonal are zero,

upper numbers could be zero.

One lower one upper corner diagonal, lower triangular, call one of the top triangle.

Any zero vector x matrix

When you hit, giving the zero vector matrix.

The only way to achieve this, for all x

The only way to provide it is not zero, all of that matrix components.

Similarly,

unit matrix on again when you multiply all x with x data.

So a neutral matrix, does no effect on the x.

And on a diagonal matrix that can provide it

We see easily that there should be reset his outside.

Refer to the matrix on a diagonal,

çarpsa others in a matrix with zero vector x,

you get the x vector again.

Namely, this means multiply the vector x will receive this vector x,

We will receive the vector inner product line consisting here of bringing in a horizontal position.

So, we hit one x1'l to reset other components will multiply.

Therefore, this product only and remain the first line x1 x1 keep.

Similarly, the second line again this xi your çarpsa brought to a horizontal position,

We see that not only had the number of X2 and all that similar.

And apart from that another matrix that is not possible,

you hit the x to give him.

This unit has a number of in how the matrix, a number that is important,

If you multiply a number by any count, still gives that number.

Its equivalent in the generalized to the matrix,

This unit is supposed to be a square matrix and the matrix.

Otherwise, this example the

you may receive a seven-line vector,

When you hit, you will not find a number of opposing this seventh row.

Let's have a zero koysam to bring it,

You may not remove x7'y the end somehow.

Because this is rectangular, though, will put zeros

What you put alternatively, you can not disconnect here x7'y of.

So you try to produce this, the matrix you want to be the identity matrix

we find easily that it was impossible to be rectangular.

Jordan also has a type called a matrix.

This looks like a diagonal matrix.

There are some numbers on the diagonal.

Some sporadic might be zero.

But there are also a number of second-degree diagonal above the diagonal.

Some of them may be zero.

All'd already diagonal matrix becomes zero.

This has a special structure.

But it is possible to define the matrix of power when the square matrices.

Because the second matrix to multiply matrix itself

The first column of the matrix, I'm sorry I said the opposite,

The number of items in the second matrix row

It has come to equal the number of columns of the first matrix.

Otherwise, you çarpamazs.

In this matrix, that is to say, however, lines and

column of the forces in case of equality,

for example, a square, cube, you can define the n'yinc forces.

We define zero-force units as well Matira.

Let's make a concrete examples.

Matrix matrix we get this get here.

Multiply the square will say to take.

Here we are writing, we are writing to another; Take the first column,

We brought crashing horizontally.

Yet because we understand immediately see that the frame

both in the second column because the

Number two elements, when we want to hit this A'yl turn it horizontally,

the same should be equal to the number of columns.

We see here also the frame without the frame can not be described.

Hit by this square is easily obtained.

If inverse matrix, the opposite we received here.

This can make the easy supply.

Minus combine to A'yl to minus one and we hit, namely the right and left

When we hit, we easily that the unit matrix.

See it easier to watch here.

We wrote a negative one as the first matrix.

And we claim that we found here,

It found that has not been mentioned yet, but how,

When we multiply the left and right in such a matrix,

We bring the landscape to take this column as you see, one half

multiplied by one plus one gives the product of a split is still one of the two.

Refer to give one here.

If we multiply this first column of the second line,

one half again, this time minus one divided by zero gives two.

As you can see zero.

Similarly also,

When we make a split two horizontal take this second column,

minus one half, when you hit zero, and again in the second row,

There is one more minus two minus one over the columns of the second element,

one half plus one half it is still giving it a.

As you can see, the unit formed matrix case.

If we multiply in reverse, we see again that the matrix of this unit.

Left or right is snapped in non-square matrix

need not be defined but they are both equal.

Even if you have one, you may not have the other.

Let's say how the five numbers in a number of you again.

Five minus the square we'd like to find a second force, there are two paths before us.

Five eksinc of negative forces to locate the second, we get five of the frame 25.

We take the opposite 1/25.

Or to one, when we take the opposite five, a division of five.

When we get one over five frames in 1/25.

As you can see, it's the equivalent number in there.

You can not make it in other public squares in the matrix, the matrix but it

You can provide.

Or, say it again, giving it an equivalent thereof, or it can come a

multiplied by the negative forces could be described that way.

A minus minus n'yinc first force with force

the mean minus the bump comes less equivalent to the first force.

Here, too, we will deliver.

In contrast to the negative we can find two of the square.

We stood on the square A'yl to find here.

Chance here two, get two easy that he had chosen.

The converse is also one half, on a split of this position will be two.

A second force means negative, it can be calculated in two ways.

We found the square before.

Square we found here.

We take the opposite.

A diagonal split over two interest divided by a dilemma.

Merger or minus one minus I have found here, we will take a negative hit again.

See here seems more complicated.

But all of this mess again collected a split two, a split will be two.

We translate this as taking the first horizontal column again.

one half, hit by a divide by two,

divided by a trailing four plus one over the coming two quarters.

When we hit the second row have the same numbers, but one of a cross hair

because it's missing a trailing slash them we find a four minus four to zero.

Again, the second column to the first

When we crossed the line we see that zero come easily.

The product of the same number, one half plus one divided by two,

One of the pros and cons of taking a slash each other's dilemma.

As a last item, the second column vector of two,

We stood with the first row of the first matrix.

As you can see there is a split in two.

There's a negative, there's a negative here.

He turns to two plus a split.

Divided by a trailing four plus four, turns out to be a split in two.

This also in the square, we saw here, a reverse split of the two,

It shows that there is a divide dilemma.

Already this we multiply the two, we see easily that one of the two

When we hit a reset split a two zeros.

Similarly here, when we turn to take the second column zero

zero and zero reset

multiplied by two and a split comes as a two multiplied.

Now I want to pause here.

As these concepts through a bit out of this very entry level,

We will make progress if you understand these examples.

Here, as an introduction to determinants, two equations with two unknowns

and learn what we can from the three equations with three unknowns, we'll look at him.

Determinants of us have learned how this concept naturally

appear to show that, for more general matrices

us to develop some intuition for the calculation

and generalize them to help us.