document.write( "Question 1164529: Suppose A is a square matrix. Prove that there is a symmetric matrix B and a skew-symmetric matrix C such that A = B + C. In other words, any square matrix can be decomposed into a symmetric matrix and a skew-symmetric matrix (Proof Technique DC). \n" ); document.write( "

Algebra.Com's Answer #788930 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "For any square matrix A,   \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    A =  + .     (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In this decomposition,   is the matrix A \"transposed\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In the decomposition (1), the matrix  B =   is s symmetric matrix,  and  the matrix  C = \r\n" );
document.write( "\r\n" );
document.write( "is the skew-symmetric matrix.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So (1) provides a required decomposition.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );