Now, let's Sturdee.
The bras get notations in left and the algebraic operations performed with it.
Don't get to Victor, that you see is the column vector consisting of elements, even a two a three
N.
These are complex numbers.
And as you know, complex numbers have an imaginary part.
Victor, is the transport of the complex conjugate of gate victor, hence it becomes the rule vector
like this.
And we know what is a complex conjugate.
It's just a reversal of the sign containing imaginary part.
Here are some algebraic operations on the bracket vector's.
Let's see.
First, we try and head to get Vector's.
Get Victor A plus, get Victor B, and we get get Victor C..
So let's say the key to here is a column, Victor, consisting of complex numbers.
A-one, a two and a three.
B consists of B1, B2 and B3, so lation is quite simple, you just added element to element.
So the final vector would be even plus B1, A2, plus B2, A3, plus B3.
Then let's check out the multiplication back over here.
Let's say Alpha is a scalar quantity, which is.
A real number or maybe a complex number, and you multiply that with a get victory.
It's pretty simple, the final great victor or the output that Victor would be Alpha, even Alpha,
A2, Alpha A3.
Now, this is a very important operation called inner product.
That is when you basically have a and get Victor.
Then how basically.
You multiply them, so, you know, Abramovic is nothing but the transport of the complex conjugate
of the jet vector.
So let's say.
You have a Brabec to and you do that in a product of that, with me, it looks like this.
Even conjugate it to conjugate A3 conjugate multiplied by a column vector, B1, B2, B3, it's very
much similar to the matrix multiplication that we saw earlier.
So the answer is even conjugate B1 plus A2 conjugate B2 plus A3 conjugate B3.
Simbi, you do the inner product of Ravich to be with get vector in.
And the result is B1 conjugate multiplied by A1 plus B2 conjugate A2, plus B3 conjugate A3.
So finally we derive this.
When we have an inner product of Brabec that Evett get Victor B, it's nothing but.
The inner product of gravity to be with character, if it's complex conjugate.
Now, this is the Matrix Algebra with the character vector, so let's say you have a Matrix M1 one,
M1 two, M2 one, M2 two and the cat vector A1 one, A1 to.
Basically, how we multiply two matrices, you are multiplying this in the same way.
And finally you get a column vector, which we call it a C, right.
So the column vector is nothing but.
And one, one even plus and one to.
Even to.
An extra.
M2 one, even one plus M2 to to.
Yeah, so it basically gives you a column vector.
Now, in crisis, there is a concept of unitary matrix.
So what is called a unitary matrix, let's say you multiply a matrix with it's a joint, you guys know
what is in a joint.
Right, we studied this concept in the Mattresses chapter, so a matrix multiplied by it's a joint.
If you get an identity matrix, that's a unitary matrix.