Transpose of a Matrix

Definition:

The transpose of a m by n matrix is defined to be a n by m matrix that results from interchanging the rows and columns of the matrix. The transpose of a matrix is designated by the superscript T or " ' ".  The matrix  and the transpose of a matrix  are as follows.

Examples;

Properties of the Transpose

The following properties are valid for the transpose;

• The transpose of the transpose of a matrix is the matrix itself:  
• The transpose of a matrix times a scalar (k) is equal to the constant times the transpose of the matrix:
• The transpose of the sum of two matrices is equivalent to the sum of their transposes:  
• The transpose of the product of two matrices is equivalent to the product of their transposes in reversed order:   
• The same is true for the product of multiple matrices:  

Symmetric Matrix

A square matrix ,  is defined as symmetric if the matrix is equal to its transpose or: C=C^T

i.e., for each element of

Example of a Symmetric Matrix

Note: The elements located symetrically with respect to the principal diagonal are equal.

Skew-Symmetric Matrix.

A square matrix is skew-symmetric if its negative is equal to its transpose.

-=^T , i.e., for components of , c_ij=-c_ji

Example of Skew-Symmetric Matrix

=

Note: The elements located symmetrically with respect to the principal diagonal (consisting of zeros) are equal in magnitude and the negative of each other.