Content

• Steps
• Part 1 of 3: Transpose the Matrix
• Part 2 of 3: Transposition Properties
• Part 3 of 3: Hermitian conjugate matrix with complex elements
• Tips

If you learn how to transpose matrices, you will have a better understanding of their structure. You may already know about square matrices and their symmetry, which will help you master transposition. Among other things, transposition helps transform vectors into matrix form and find vector products. When working with complex matrices, Hermitian-conjugate (conjugate-transpose) matrices can help you solve a variety of problems.

Steps

Part 1 of 3: Transpose the Matrix

1. 1 Take any matrix. Any matrix can be transposed, regardless of the number of rows and columns. Most often, you have to transpose square matrices that have the same number of rows and columns, so for simplicity, consider the following matrix as an example:
   • the matrix A =
     1  2  3
     4  5  6
     7  8  9
2. 2 Imagine the first row of a direct matrix as the first column of the transposed matrix. Just write the first line as a column:
   • transposed matrix = A
   • first column of matrix A:
     1
     2
     3
3. 3 Do the same for the rest of the lines. The second row of the original matrix will become the second column of the transposed matrix. Translate all rows to columns:
   • A =
     1  4  7
     2  5  8
     3  6  9
4. 4 Try to transpose a non-square matrix. Any rectangular matrix can be transposed in the same way. Just write the first line as the first column, the second line as the second column, and so on. In the example below, each row of the original matrix is ​​marked with its own color to make it clearer how it is transformed when transposed:
   • the matrix Z =
     4  7  2  1
     3  9  8  6
   • the matrix Z =
     4  3
     7  9
     2  8
     1  6
5. 5 Let us express the transposition in the form of a mathematical notation. Although the idea of ​​transposition is very simple, it is best to write it down as a strict formula. Matrix notation does not require any special terms:
   • Suppose given a matrix B consisting of m x n elements (m rows and n columns), then the transposed matrix B is a set of n x m elements (n rows and m columns).
   • For each element b_xy (line x and column y) of the matrix B in the matrix B there is an equivalent element b_yx (line y and column x).

Part 2 of 3: Transposition Properties

1. 1 (M = M. After double transposition, the original matrix is ​​obtained. This is pretty obvious, since when you re-transpose, you change the rows and columns again, resulting in the original matrix.
2. 2 Mirror the matrix around the main diagonal. Square matrices can be "flipped" relative to the main diagonal. Moreover, the elements along the main diagonal (from a₁₁ to the bottom-right corner of the matrix) remain in place, and the rest of the elements move to the other side of this diagonal and remain at the same distance from it.
   • If you find it difficult to imagine this method, take a piece of paper and draw a 4x4 matrix. Then rearrange its side elements relative to the main diagonal. At the same time, trace the elements a₁₄ and a₄₁... When transposed, they must be swapped like other pairs of side elements.
3. 3 Transpose the symmetrical matrix. The elements of such a matrix are symmetric about the main diagonal. If you do the above operation and "flip" the symmetric matrix, it will not change. All elements will change to similar ones. In fact, this is the standard way to determine if a given matrix is ​​symmetric. If the equality A = A holds, then the matrix A is symmetric.

Part 3 of 3: Hermitian conjugate matrix with complex elements

1. 1 Consider a complex matrix. The elements of a complex matrix are composed of real and imaginary parts. Such a matrix can also be transposed, although in most practical applications conjugate-transposed or Hermitian-conjugate matrices are used.
   • Let a matrix C =
     2+i     3-2i
     0+i     5+0i
2. 2 Replace the elements with complex conjugate numbers. In the operation of complex conjugation, the real part remains the same, and the imaginary part changes its sign to the opposite. Let's do this with all four elements of the matrix.
   • we find the complex conjugate matrix C * =
     2-i     3+2i
     0-i     5-0i
3. 3 We transpose the resulting matrix. Take the found complex conjugate matrix and simply transpose it. As a result, we get a conjugate-transposed (Hermitian-conjugate) matrix.
   • the conjugate-transposed matrix C =
     2-i        0-i
     3+2i     5-0i

Tips

• In this article, the transposed matrix relative to the matrix A is denoted as A. There is also the notation A 'or Ã.
• In this article, the Hermitian-conjugate matrix with respect to the matrix A is denoted as A, which is a common notation in linear algebra. In quantum mechanics, the notation A is often used.Sometimes a Hermitian conjugate matrix is ​​written in the form A *, but it is better to avoid this notation, since it is also used to write a complex conjugate matrix.