Linear Algebra - Matrix Symitry

[JeeebeZ]

Homework Statement

A matrix A is skew-symmetric if A[SUP]T[/SUP] = -A. Write the matrix
B below as the sum of a symmetric matrix and a skew-symmetric matrix.
B =  a b c
     d e f
     g h i

The Attempt at a Solution

So I'm Pretty sure that the
Symetric Matrix = B + B^T
Skew Symetric Martix = B - B^T

So B+B^T+ B - B^T should equal B but I get 2 B.

So B = 1/2((B+B^T) + (B - B^T))

B[SUP]T[/SUP] = a d g
     b e h
     c f i

Sym = B + B[SUP]T[/SUP]
    =  2a b+d c+g
      d+b  2e f+h
      g+c h+f  2i

Skew Sym = B - B[SUP]T[/SUP]
    =  0  b-d c-g
      d-b  0  f-h
      g-c h-f  0

So If you add Those... You get

2a 2b 2c = 2B
2d 2e 2f
2g 2h 2i

Then divide by 2 and I'm done right?

 

[SammyS]

Homework Statement

A matrix A is skew-symmetric if A[SUP]T[/SUP] = -A. Write the matrix
B below as the sum of a symmetric matrix and a skew-symmetric matrix.
B =  a b c
     d e f
     g h i

The Attempt at a Solution

So I'm Pretty sure that the
Symetric Matrix = B + B^T
Skew Symetric Martix = B - B^T

So B+B^T+ B - B^T should equal B but I get 2 B.

So B = 1/2((B+B^T) + (B - B^T))

B[SUP]T[/SUP] = a d g
     b e h
     c f i

Sym = B + B[SUP]T[/SUP]
    =  2a b+d c+g
      d+b  2e f+h
      g+c h+f  2i

Skew Sym = B - B[SUP]T[/SUP]
    =  0  b-d c-g
      d-b  0  f-h
      g-c h-f  0

So If you add Those... You get

2a 2b 2c = 2B
2d 2e 2f
2g 2h 2i

Then divide by 2 and I'm done right?

Actually, you're pretty close to getting a correct result.

The symmetric matrix you need is (1/2)(B + B^T), the skew-symmetric matrix is (1/2)(B - B^T) .