The transpose AT of a matrix A is the exchange of rows and columns of the matrix.
|A = [||1||1||-2||], B = [||-1||0||1||], C = [||1||2||], D = [||0||1||], X = [||a||b||c||]|
|AT = [||1||3||], BT = [||-1||-2||], CT =||
|, DT =||
|, XT = [||a||x||]|
In general, if we write a matrix in a compact form A = (aij), then AT = (aji).
|A = [||a11||a12||...||a1n||] ⇒ AT = [||a11||a21||...||am1||]|
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.
The following are the general forms of 2 by 2 and 3 by 3 skew-symmetric matrices.
The transpose has the following properties. The linearity and involutive property are quite obvious. The product property is not so obvious.
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.
|scalar multiplication||scalar multiplication|
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.