If we further simplify this equation or her mission matrix multiplied by an eigenvector gives and eigenvalue
multiplied by the same eigenvector.
So here there could be two or more eigenvectors orthogonal to each other for her mission matrix, which
gives an eigenvalue for that vector.
So the eigenvalue would change based on the vector for a different eigenvector, there could be different
eigenvalue.
Now, what do we literally mean by orthogonal at the inside?
We know that orthogonal means perpendicular, but let's study in the form of an equation.
Let's say we got to Eigenvectors eEye and a game of automation matrix like this.
So if you take the inner product of AA and AK, it's equal to one if I is equal to.
It's equal to zero if I is not equal to.
If we do the inner product of them, we get one.
If I is equal to Cain, seems like the same vectors and we get zero if I is not equal to.
So one commissioned matrix.
With two or more eigenvectors with respect to iGun values, so the first thing is we know that.
A B, the inner product of A would be if you do the conjugate of that, it becomes B, A being a product
of B with A..
So if you take these two equations over here, Lambda is a real quantity.
We call it as an eigenvalue.
Now, what exactly do we mean by this equation?
It means H.
Is the commission operator operating on the target vector, which results in a vector whose inner product
we do with B.
So if I replace it A with lambda a E now, lambda being real can come out and we can do the inner product
of B and A.
Now let's look at the fourth state of polarization and figure it out.
Why it represented as column vectors like this, how these column vectors actually came about.
Here we see the horizontal state with vector one zero.
We can ride this as one times X where X is the horizontal component and zero times Y where Y is the
vertical component.
And if you recall, it has eigenvalue of one.
And the length of the vector is one that is one square plus zero square equals one.
Similarly, if we take the vertical state of polarization with Vector 01, we represent this as zero
times X plus one times Y where X is the horizontal component.
And why is the vertical component and the length of the vector is one that is zero square plus one square,
which is equal to one.
Now let's take the forward slash state of polarization, which is represented as one by two, one by
two.
So we can write this as one zero two X plus one by two to Y.
With an angle of 45 degree and the length of the vector is one birute, two squared plus one by to square
is equal to one.
Let us come to the fourth state of polarization, which is in backward slash state.
It's represented as one by two to minus one by two, which is equal to one by to X, minus one Birute
to Y, and the length of vector is again one that is one Birute two squared plus.
Minus one by two to square, which is equal to one.
Now, a very important piece of information, let's study probability in these four states how we can
discover probability from these four states, this first horizontal state has a probability of one square
is equal to one that is 100 percent passing through the horizontal factor and probability of zero square.
Which is equal to zero on a zero percent passing through the vertical filter in the second state, which
is the vertical state, probability of passing through a horizontal filter is zero square, which is
equal to zero.
That is zero percent chance of passing through.
While one square is equal to one that is one hundred percent chance of passing through the vertical
state of the vertical freedom, similarly for forward slash state, it is one by to square.
That is one by two equals point five, which is 50 percent chance of passing through the horizontal
filter and similarly, 50 percent chance of passing through the vertical filter.
And this applies to the backward slash state as well.
Let's see if we got Victor at an angle like this now teda his to the horizontal axis with lenth one,
then the vector we need to represent as the angle theta can be represented as.
To probably the amplitudes, alpha and beta, which can be written as Alpha X plus Betawi.
So here Alpha is the X coordinate and can be written this cost theta.
You know what is Costa Mesa on trigonometry?
It's a descend upon hypotenuse.
And Beita is the Vicodin, it readiness Sindy to.
That is opposite upon hypotenuse, so it can be written as the gateway to Peter, which is equal to
Constitución data, so the probability of any photon blazed into state to pass over a horizontal state
is greater and over vertical state is sine granted.
Now, this becomes very important to study so that we can get to know how the photon speed and the polarizer
are working together.
If I draw a vector having an angle peeter with x axis and draw an orthogonal vector, you know what
is also going to create ninety degree do it like this, then we can see that this vector also makes
angle theta with y axis.
So this Victor, looks like the backward slash state.
And we can represent it as a vector minus 90 to Kostina.
And this week looks like the forward slash state, which has vectored like Kosti to find detail.
So now if I prepare a fortune, stayed in backward direction and pass it through a forward slash polarizer,
polarizer is nothing but a filter.
Yeah, the photon passing through a filter.
Then the probability amplitude is the inner product of both the vectors, which is zero.
Similarly, if both the state and polarizer are in the same condition or state like this.
We're both the 41st and Polarizer Odyn forward slash state.
The probability amplitude is one and the probability is one square is equal to one.
Yeah, so the chance of passing the photon through this polarizer is 100 percent.
The probability is a hundred percent.
Finally, a big question is answered.
Now, you know, you had a question that how are the values of information?
Matrix decided.
Here is your answer.
It's cost two to sign to Peter.
Signed to Teda minus cost to the.
Josh.