Review of Symmetric and Skew-symmetric Matrices

Transpose of a matrix is the matrix obtained by changing its rows into corresponding columns or columns into corresponding rows. The transpose of a m × n matrix is a n × m matrix that results by interchanging the rows and columns of the matrix. The transpose of a matrix is written with the superscript T or ‘.

 

 

 

Symmetric Matrix

A square matrix is defined as symmetric if it is equal to its transpose, i.e. A = A^T. for each element of A, a_ij = a_ji.

Example,

 

 

 

 

As A = A^T. A is a symmetric matrix,

Skew-symmetric Matrix

A square matrix can be a skew-symmetric matrix if its negative is equal to its transpose i.e. –A = A^T. For every element of A, , a_ij = -a_ji.

Example,

 

 

 

 

 

 

As –A = A^T, A is a skew-symmetric matrix.

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