Fix n N For an n times n matrix A Aij1 lessthanorequalto i

Fix n N. For an n times n matrix A = (A_ij)1 lessthanorequalto i, j lessthanorequalto n, the transpose of A is the n times n matrix A^T defined by (A^T)_ij: = A_ij for 1 lessthanorequalto i, j lessthanorequalto n. An n times n matrix A is called symmetric if A^T = A. Further, A is called skew-symmetric if A^T = -A. Let V be the vector space of all n times n matrices. Define V_+:= {A V \\ A^T = A} and V_:= {A V \\ A^T = -A}, so V_+ is the set of all symmetric n times n matrices, and is the set of all skew-symmetric n times n matrices. (a) Prove that V_+ and V_ are subspaces of V. (b) Prove that the vector space of all n times n matrices is spanned by V_+ and V_;i.e., prove that V = V_+ + V-.

Solution

a)

1. Let, X,Y be in V+

(X+Y)^T=X^T+Y^T=X+Y

Hence, X+Y is in V+

2. Let, X be in V+ and c be a scalar

(cX)^T=cX^T=cX

Hence, V+ is a subspace

1. Let, X,Y be in V-

(X+Y)^T=X^T+Y^T=-X-Y

Hence, X+Y is in V-

2. Let, X be in V- and c be a scalar

(cX)^T=cX^T=-cX

Hence, V- is a subspace

b)

Let, A be any nxn matrix

SO, A can be written as

A=(A+A^T)/2+(A-A^T)/2

(A+A^T)/2=(A+A^T)/2

(A-A^T)/2=(A^T-A^T)/2=-(A-A^T)/2

HEnce, any nxn matrix can be written as sum of a symmetric an skew symmetric matrix

HEnce proved