 In this video, I want to introduce another operation for matrices. This was pretty simple. It's called the transpose of matrix the operation We call transposition. So imagine we have a matrix given by the formula a equals The matrix whose ith j position is a ij. Well, the transpose of a which will denote a to the superscript t We're gonna so if a is an in by in matrix the transpose is gonna be an in by in matrix Notice how that's swapped around and the entries of a transpose are going to be a j i You'll notice that we swap the index around and this is what the transpose does the transpose is going to swap the roles of columns and rows The rows of a are going to become the columns of a transpose and the columns of a are going to become the Rows of a transpose so rows and columns are going to swap around So an example of that take the matrix a right here a Trans so a here is a 2 by 3 matrix and its transpose is gonna be a 3 by 2 matrix the first The first row of a becomes the first column of a transpose and then the second row of a Becomes the second column of a transpose so the row becomes columns and columns become rows Like so we did the same thing for B B is likewise a 2 by 3 matrix its transpose will be 3 by 2 So we take the first row of B that becomes the first column of B transpose We take the second row of B and that becomes The second column of B transpose or if you prefer we take the first column of B. That's the first Row of B transpose we take the second column of B that becomes the second row of B transpose And we take the third column of B and that becomes the third row for B That's all there is to it if you take a matrix C for example its first row 2 negative 3 becomes the first column of C transpose and then the Second row of C becomes the second row of the second column of C transpose And so this 2 by 2 matrix will turn into a 2 by 2 matrix Another thing I want to mention is that when you compare a matrix with its transpose The main diagonal doesn't change So like when you look at a here you get one and five is the main diagonal Same thing for B here You get the main diagonal one and five for C your main diagonal is two and one that stays the same So some people like to think of it as follow when you take the transpose of a matrix You're reflecting the matrix across this main diagonal That's a good perspective to take here and transposition is no more difficult than what we see right here Now there I have to I have to add one little caveat to that statement right there in the case of complex matrices So matrices over the complex field it turns out that the transposition Operation we just introduced is insufficient. So we have to propose an alternative for transposes again in the case for Complex matrices, this will be more clear in chapter 5 why we need this alternative But for complex matrices we introduced the notion of the conjugate transpose So again a is an in-by-in matrix the conjugate transpose, which will denote this as a star. This will be the conjugate of a Transposed where transpose still has the same meaning you're going to reflect across the main diagonal But this line over the matrix. This is our this is the complex conjugate operation This is the operation where if you take a complex number a plus b you draw a line over it You get the conjugate you're going to switch the sign of the imaginary Component and so this conjugate transpose will be a big role for complex matrices So one thing I want to mention is that when we're discussing transposes of matrices Whenever that matrix happens to be complex always always always with no exceptions Substitute the conjugate transpose in for the standard transpose So let's take two complex matrices ones two by three That's a and then B right here is three by three. So the conjugate transpose a star We're going to take the first row of a we're going to flip it That becomes now the first column of a star But notice we took conjugates here that a our one minus two becomes one sorry one minus two i becomes one plus two i and then Three plus five i becomes three minus five i you switch the sign I'm a negative a negative two i becomes a positive two i and a plus five i becomes a negative five i When you take the conjugate of six that actually stays six because you can think of six right here six is Six plus zero i so it's conjugate would be six minus zero i which is still to six when you take the conjugate of a Real number it doesn't do anything and then for the second row The second row of a becomes the second column of a star, but you take conjugates again So negative two i which is a purely imaginary number becomes positive two i the conjugate of zero is zero because it's a real number And then the conjugate of i becomes negative i so you switch the sign of all the imaginary parts in addition to conjugating For B if we want to compute B star we could we could actually think of it in terms of columns Right so the first column of B becomes the first row of B star But take conjugates two minus three i becomes two plus three i negative i becomes i and zero becomes zero the second column of B becomes the second row of B star with also Taking conjugates zero's conjugate is zero force conjugate is four because those are both real numbers But two minus two i becomes two plus two i and then taking the third Column that will become the third row of B star and then make sure you also conjugate the most common mistake with these complex matrices As we forget to do the conjugate the conjugate of two i becomes a negative two i And the conjugate of one plus two i becomes one minus two i then the conjugate of six is again six It's just a real number and so that's how we do conjugates and and That's how we do transpositions for real matrices and we do we do Conjugate transposes for complex matrices if we ever have to do transpositions over, you know Any other field we'll just take the regular transposition this conjugate transposes only going to be defined in the for this series for complex matrices