The transpose of a matrix is the matrix found by switching rows for columns: more formally, if A = [a_{ij}]: m*n then A^{T} = [a_{ji}]: n*m (that is, the entry in the i^{th} row, j^{th} column of A becomes the entry in the j^{th} row, i^{th} column of A^{T}).

The following properties hold for the transpose:

(i) (A+B)^{T} = A^{T} + B^{T} (the transpose of the sum is the sum of the transposes)

(ii) (AB)^{T} =B^{T}A^{T}

(iii)A^{TT} = A

The third property is trivially true, and the first is fairly obvious when you consider the elements:

(A + B)^{T} = ( [a_{ij}]: m*n + [b_{ij}]: m*n)^{T}

= ( (a_{ij} + b_{ij}): m*n )^{T}

= [a_{ji} + b_{ji}]: n*m

= [a_{ji}]: n*m + [b_{ji}]: n*m

= A^{T} + B^{T}

However, the second requires some more thought. Consider the i,j^{th} element of (AB)^{T}, which equals the j,i^{th} element of AB (by definition of transpose). This is found by taking the scalar (dot) product of the j^{th} row of A and the i^{th} column of B (usual process of matrix multiplication). However, this is identical to the scalar product of the j^{th} column of A^{T} and the i^{th} row of B^{T}, namely (B^{T}A^{T})_{ij}. So property (ii) holds. Note that the order matters- it may not even be possible to define the product A^{T}B^{T} but given AB (ie A: m*n, B: n*p to give an m*p matrix) can be calculated, B^{T}A^{T} (p*n multiplying n*m to give p*m as expected) can be.

A matrix is described as *symmetric* if A^{T}=A: switching the rows and columns has no effect. Similarly, if A^{T}= -A
(exchanging rows for columns effectively switches signs) then A is described as *skew-symmetric*. If A is a square (i.e. m=n, same number of rows as columns) matrix, then A + A^{T} is always symmetric:

(A + A^{T})^{T} = A^{T} + A^{TT} by property (i)

= A^{T} + A by property (iii)

=(A + A^{T}) so the definition of symmetric is met.

and by similar logic, (A - A^{T}) can be shown to be skew-symmetric. Thus using the properties of transpose matrices, all square matrices can be expressed uniquely in terms of a sum of a symmetric matrix and a skew-symmetric matrix.

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