### Transpose

The transpose AT of a matrix A is the exchange of rows and columns of the matrix.

Example Let

 A = [ 1 1 -2 ], B = [ -1 0 1 ], C = [ 1 2 ], D = [ 0 1 ], X = [ a b c ] 3 0 1 -2 1 1 3 4 1 2 x y z

Then

AT = [ 1 3 ], BT = [ -1 -2 ], CT =
 [ 1 3 ] 2 4
, DT =
 [ 0 1 ] 1 2
, XT = [ a x ]
1 0 0 1 b y
-2 1 1 1 c z

In general, if we write a matrix in a compact form A = (aij), then AT = (aji).

 A = [ a11 a12 ... a1n ] ⇒ AT = [ a11 a21 ... am1 ] a21 a22 ... a2n a12 a22 ... am2 : : : : : : am1 am2 ... amn a1n a2n ... amn

Moreover, if A is an m by n matrix, then AT is an n by m matrix. The rows of A are the columns of AT, and the columns of A are the rows of AT.

A matrix A is called symmetric if AT = A. It is called skew-symmetric if AT = - A. The matrix D in the example above is symmetric. Clearly, a symmetric (or skew-symmetric) matrix must be a square matrix.

Example The following are the general forms of 2 by 2 and 3 by 3 symmetric matrices.

 [ a b ] b c
 [ a b c ] b d e c e f

The following are the general forms of 2 by 2 and 3 by 3 skew-symmetric matrices.

 [ 0 a ] -a 0
 [ 0 a b ] -a 0 c -b -c 0

The transpose has the following properties. The linearity and involutive property are quite obvious. The product property is not so obvious.

linearity
(A + B)T = AT + BT, (cA)T = cAT
product
(AB)T = BTAT
involutive
(AT)T = A
inverse
(A-1)T = (AT)-1

The properties can also be explained by the interpretation of the transpose in terms of the linear transformations. In the equivalence between matrices and linear transformations, we would like to find the answer to 4? in the following table.

 linear transformation matrix composition multiplication addition addition scalar multiplication scalar multiplication inverse inverse 4? transpose

The answer is the dual of linear transformations. The linearity and the product properties are quite easy to see from the transformation viewpoint. The interpretation of the involutive property is somewhat complicated and can be found here.