# Index notation of matrix tranpose

• birulami
In summary, Zee explains the definition of the transpose and emphasizes the importance of the left-to-right order of indices in a matrix representation. This is demonstrated through examples using upper and lower indices. The left-to-right order determines the row and column indices in a matrix. Zee also addresses a potential typo and clarifies the meaning of a slightly shifted index.

#### birulami

Zee writes in Einstein Gravity in a nutshell page 186

"let us define the transpose by ##(\Lambda^T)_\sigma^\mu = \Lambda_\sigma^\mu##"

and even emphasizes the position of the indexes. Yes, they are not exchanged! This must be a typo, right?

No, what he says is perfectly fine: he writes ##\Lambda^{\mu}{}{}_{\sigma} = (\Lambda^T)_{\sigma}{}{}^{\mu}## which is not what you wrote above.

And you are telling me now that this glitch in the typography of the slightly out of place indexes is the crucial point?

Oh my, I thought I understood why we have upper and lower indexes? But what is the meaning of an index slighly shifted to the right?

birulami said:
And you are telling me now that this glitch in the typography of the slightly out of place indexes is the crucial point?

Oh my, I thought I understood why we have upper and lower indexes? But what is the meaning of an index slighly shifted to the right?

The left-to-right order of the indices matters: in a matrix representation the first index is the row index and the second index is the column index.

$$A_\lambda{}^\mu = g_{\lambda\nu} \, A^{\nu\mu} = g_{\lambda\nu} \, g^{\mu\sigma} \, A^\nu{}_\sigma$$

No, this is not a typo. The index notation of matrix transpose is defined as follows:
The transpose of a matrix A is denoted as A^T, and its elements are given by (A^T)_ij = A_ji. In other words, the rows and columns of the matrix are flipped, and the indexes are not exchanged. This is the standard definition of transpose in mathematics and is consistent with the notation used in physics, as shown in Zee's book. So, there is no typo here, and this notation is correct. It is important to note that this notation is different from the notation used in some programming languages, where the indexes are indeed exchanged. As a scientist, it is crucial to be aware of these notational differences to avoid confusion and ensure accurate communication in our work.

## 1. What is the purpose of using index notation in matrix transpose?

Index notation allows us to represent and manipulate matrices in a concise and efficient manner. It also allows us to easily perform operations such as matrix transpose without having to write out the entire matrix.

## 2. How is index notation used to represent matrix transpose?

In index notation, the transpose of a matrix A with dimensions m x n is represented as AT and can be written as (Aij)T = Aji. This means that the element in row i and column j of matrix A will become the element in row j and column i in its transpose.

## 3. Can index notation be used for matrices with more than two dimensions?

Yes, index notation can be extended to higher dimensional matrices. For example, for a 3-dimensional matrix A with dimensions m x n x p, the transpose would be represented as AT and can be written as (Aijk)T = Akji. This means that the element in row i, column j, and depth k of matrix A will become the element in row k, column j, and depth i in its transpose.

## 4. How is index notation helpful in performing operations on matrices?

Index notation allows us to perform operations on matrices in a more organized and structured way. It also makes it easier to keep track of the indices and avoid errors. This notation is especially useful when dealing with large matrices or performing complex operations.

## 5. Are there any rules or properties for index notation in matrix transpose?

Yes, there are a few important properties to keep in mind when using index notation for matrix transpose. These include the commutative property (ATT = A), the distributive property (A + B)T = AT + BT, and the associative property (AB)T = BTAT. Additionally, the transpose of the inverse of a matrix is equal to the inverse of the transpose (A-1)T = (AT)-1.