The structure of matrices can be understood using matrix transposes.Squareness and symmetry are features you already know about matrices.Transposition can be used to express the products of the vectors.If you're dealing with complex matrices, the concept of a conjugate transpose will help you through many problems.
Step 1: Pick a matrix and start.
No matter how many rows and columns the matrix has, you can change it.A simple square matrix with an equal number of rows and columns is an example.
Step 2: The first row of the matrix can be turned into the first column.
Re write row one of the matrix as a column.
Step 3: Continue for the remaining rows.
The second row of the original matrix is the second column.Continue this pattern until you turn every row into a column.
Step 4: A non- square matrix is a good place to practice.
The same is true for a non- square matrix.The first, second, and so forth rows are changed.There is an example with color-coding to show you where the elements end up.
Step 5: It is possible to express the transposition mathematically.
It's nice to be able to describe the concept in mathematics.If matrix B is an m x n matrix (m rows and n columns), it's an n x m matrix.The matrix B has an equal element at byx for each element bxy.
Step 6: M is for me.
The original matrix has a transpose.The only thing you're doing is changing the rows and columns.You're back where you started if you switch them again.
Step 7: Over the main diagonal, flip the square matrices.
The matrix flips over the main diagonal.The elements in a diagonal line from element a11 to the bottom right corner will not change.The elements will move across the diagonal and end up on the opposite side.Draw a matrix on a piece of paper if you can't see it.The fold is over the diagonal.See how elements a14 and a41 are connected.They trade places in the transpose and each other pair touches when folded.
Step 8: The matrix is a symmetric one.
There is a matrix across the diagonal.We can see that nothing has changed if we use the "flip" or "fold" description.The element pairs that trade places were the same.This is the standard way to define a matrix.matrix A is symmetric if it is A.
Step 9: Start with a large matrix.
A complex matrix has elements that are real and imaginary.The conjugate transpose is the most practical calculation for these matrices.Matrix C is a matrix.
Step 10: Take the complex conjugate.
The complex conjugate does not change the real components.This operation should be performed for all elements of the matrix.A complex conjugate of C.
Step 11: Listen to the results.
Take an ordinary change of the result.The conjugate of the original matrix is what you end up with.There is a conjugate of C.