Fernando Revilla said:I suppose you mean a symmetric matrix different from $0.$ If $\mathcal{S}$ is the subspace of the symmetric matrices, then $\dim \mathcal{S}=\dfrac{3(3+1)}{2}=6.$ If $\dim (\mathcal{S}\cap U)=0,$ then by the Grassmann theorem $\dim (U+\mathcal{S})=6+4-0=10>9=\dim \mathbb{K}^{3\times 3}$ (contradiction). So, there exists a symmetric and non null matrix belonging to $U.$
A symmetric matrix is a square matrix where the elements above and below the main diagonal are mirror images of each other. In other words, the element at row i and column j is equal to the element at row j and column i.
To determine if a matrix is symmetric, you can check if the matrix is equal to its transpose. If the elements are the same, then the matrix is symmetric.
Symmetric matrices have many important applications in mathematics and physics. They are used to represent real-world systems such as networks, graphs, and physical systems. They also have special properties that make them easier to work with in calculations.
No, a non-square matrix cannot be symmetric. Symmetry is a property of square matrices where the number of rows and columns are equal.
In data analysis, symmetric matrices are used in methods such as principal component analysis (PCA) and cluster analysis. They are also used in machine learning algorithms to reduce the dimensionality of data and improve computational efficiency.