# Transpose Operation

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A Transpose Operation is a Linear Algebra Operator that flips a matrix or over its diagonal.

**Example(s):****Counter-Example(s):****See:**Arthur Cayley, Transpose of a Linear Map, Linear Algebra, Matrix (Mathematics).

## References

### 2020a

- (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Transpose Retrieved:2020-8-9.
- In linear algebra, the
**transpose**of a matrix is an operator which flips a matrix over its diagonal;that is, it switches the row and column indices of the matrix

**A**by producing another matrix, often denoted by**A**^{T}(among other notations).The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley.

^{[1]}

- In linear algebra, the

- ↑ Arthur Cayley (1858) "A memoir on the theory of matrices",
*Philosophical Transactions of the Royal Society of London*,**148**: 17–37. The transpose (or "transposition") is defined on page 31.

### 2020b

- (Weisstein, 2020b) ⇒ Weisstein, Eric W. “Transpose". From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Transpose.html
- QUOTE: A transpose of a doubly indexed object is the object obtained by replacing all elements $a_{ij}$ with $a_{ji}$. For a second-tensor rank tensor $a_{ij}$, the tensor transpose is simply $a_{ji}$. The matrix transpose, most commonly written $A^{T}$, is the matrix obtained by exchanging $A$'s rows and columns, and satisfies the identity $\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}$.
Unfortunately, several other notations are commonly used, as summarized in the following table. The notation $A^{T}$ is used in this work.

- QUOTE: A transpose of a doubly indexed object is the object obtained by replacing all elements $a_{ij}$ with $a_{ji}$. For a second-tensor rank tensor $a_{ij}$, the tensor transpose is simply $a_{ji}$. The matrix transpose, most commonly written $A^{T}$, is the matrix obtained by exchanging $A$'s rows and columns, and satisfies the identity