Every square matrix is the sum of a unique symmetric and a unique skew-symmetric matrix

This one is going to be alone brief post. Came across this interesting property in my advanced solids course, as well.

Consider the square matrix $\mathbf{A}$. Clearly we can write $\mathbf{A}$ like this $$\begin{align*} \mathbf{A}&=\frac{1}{2}\left(\mathbf{A}+\mathbf{A}+\mathbf{A}^T-\mathbf{A}^T\right)\\ &=\frac{1}{2}\left(\mathbf{A}+\mathbf{A}^T\right)+\frac{1}{2}\left(\mathbf{A}-\mathbf{A}^T\right) \end{align*}$$ Let $\mathbf{B}=\frac{1}{2}(\mathbf{A}+\mathbf{A}^T)$ and $\mathbf{C}=\frac{1}{2}(\mathbf{A}-\mathbf{A}^T)$. So $\mathbf{A}=\mathbf{B}+\mathbf{C}$. Based on the property that $(\mathbf{A}+\mathbf{B})^T=\mathbf{A}^T+\mathbf{B}^T$ it's easy to verify that $\mathbf{B}$ is symmetric $(\mathbf{B}=\mathbf{B}^T)$ and that $\mathbf{C}$ is skew-symmetric $(\mathbf{C}=-\mathbf{C}^T)$. Therefore a solution exists.

But, is this solution unique? Assume we have two solutions: $\mathbf{A}=\mathbf{B}_1+\mathbf{C}_1=\mathbf{B}_2+\mathbf{C}_2$ where $\mathbf{B}_i$ is symmetric and $\mathbf{C}_i$ is skew-symmetric. Since $\mathbf{B}_1+\mathbf{C}_1=\mathbf{B}_2+\mathbf{C}_2$ we know that $$\mathbf{B}_1-\mathbf{B}_2=\mathbf{C}_2-\mathbf{C}_1$$ It's easy to show that the sum of two symmetric matrices is symmetric and the sum of two skew-symmetric matrices is skew-symmetric. Therefore, the left hand side is symmetric and the right hand size is skew-symmetric. Because they are equal, it implies that they are both the zero matrix. $$\mathbf{B}_1-\mathbf{B}_2=\mathbf{C}_2-\mathbf{C}_1=\mathbf{0}$$ And lastly, we see that $$\mathbf{B}_1=\mathbf{B}_2$$ $$\mathbf{C}_1=\mathbf{C}_2$$ And really, we were looking at the same representation. Therefore, it is unique!

In the course we were really talking about how the displacement gradient $\left(\nabla\mathbf{u}\right)$ is really the sum of the infinitesimal strain tensor $\left(\mathbf{\varepsilon}\right)$ that is symmetric and the infinitesimal rotation tensor $\left(\mathbf{\omega}\right)$ that is skew-symmetric. But the same principle applies to the much easier concept of matrices.