SOLUTION: 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

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).
Answer by ikleyn(52776) (Show Source): You can put this solution on YOUR website!
.


For any square matrix A,   


    A =  + .     (1)


In this decomposition,   is the matrix A "transposed".


In the decomposition (1), the matrix  B =   is s symmetric matrix,  and  the matrix  C = 

is the skew-symmetric matrix.


So (1) provides a required decomposition.

Solved.

RELATED QUESTIONS
Suppose A is a square matrix. Prove that there is a symmetric matrix B and a... (answered by ikleyn)

2. (a) Prove that the product of a matrix and its transpose is symmetric matrix (b)... (answered by lynnlo)

Find matrix B, if A is symmetric matrix and B is skew symmetric matrix and their sum is... (answered by ikleyn)

give an example of a matrix which is normal but not symmetric ,skew symmetric or... (answered by Fombitz)

Question: Matrix A is said to be skew symmetric if A^T = -A. Show that if a matrix is... (answered by robertb)

If A is skew-symmetric matrix,show that the elements in the main diagonal are all... (answered by robertb)

let W be an nx1 matrix such that W^T*W=1.The nxn matrix H=In-2WW^T is called a... (answered by rothauserc)

A square matrix A is called symmetric if A^t = A. If B is a square matrix, then B.B^t and (answered by jim_thompson5910)

Answer each of the following as True or False justifying your answers: If S and K are... (answered by ikleyn)