Prove: Square Matrix Can Be Written As A Sum Of A Symmetric And Skew-Symmetric Matrices

Question

Let $C^{n \times n}$ be a square matrix.

Prove that $$C=\frac{1}{2}(C+C^T)+\frac{1}{2}(C-C^T)$$

What I have manage so far is:

a. Let $S$ be a Symmetric Matrix so $S=C+C^T$

b. Let $N$ be a Skew-Symmetric Matrix so $N=C-C^T$

Proof:

$S^t=[C+C^T]^T=C^T+C=S$

$N^t=[C-C^T]^T=-C^T+C=-N$

I think you mean that you want to prove that the two terms you've divided $C$ into are correctly symmetric and skew-symmetric. Also, your last equality is wrong! — Semiclassical, Jul 23 '14 at 13:33

score 28 · Accepted Answer · answered Jul 23 '14 at 13:33

28

If $A$ is a square matrix in $\mathcal M_n(\Bbb C)$ then $$A=\underbrace{\frac12(A+A^t)}_{\text{symmetric matrix}}+\underbrace{\frac12(A-A^t)}_{\text{skew-symmetric matrix}}$$

answered Jul 23 '14 at 13:33

1

Why should it be multiple by $1/2$? – gbox Jul 23 '14 at 15:16
5

Without the coefficient $\frac12$ the sum gives $2A$. – Jul 23 '14 at 15:21

score 1 · Answer 2 · answered May 10 '18 at 18:38

$$(A+A^t)_{ij} =A_{ij}+A^t_{ij} = A^t_{ji}+A_{ji} = (A^t+A)_{ji} = (A+A^t)_{ji}$$ Therefore $(A+A^t)$ is symmetric. $$(A-A^t)_{ij} =A_{ij}-A^t_{ij} = A^t_{ji}-A_{ji} = (A^t-A)_{ji} = -(A-A^t)_{ji}$$ Therefore $(A-A^t)$ is scew-symmetric. Since $$A = \frac12A+\frac12A+\frac12A^t-\frac12A^t=\frac12(A+A^t)+\frac12(A-A^t)$$ $A$ can be Written as a sum of a scew-symmetric and a symmetric Matrix.

score 0 · Answer 3 · edited May 15 '18 at 01:56

0

Using indicial notation, show that the square matrix can be expressed as the sum of a symmetric and antisymmetric matrix by the decomposition:

$$A=\frac{A+A^T}{2} + \frac{A-A^T}{2}$$

edited May 15 '18 at 01:56

Mr Pie

9,726

answered May 15 '18 at 00:47

Víctor Paniagua Villagómez

11

Prove: Square Matrix Can Be Written As A Sum Of A Symmetric And Skew-Symmetric Matrices

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