Understanding the Dimension of the Vector Space of 10 × 10 Symmetric Matrices

The concept of a vector space is fundamental in linear algebra, and symmetric matrices play a significant role in various applications such as optimization, graph theory, and physics. In this article, we will explore the dimension of the vector space of 10 × 10 symmetric matrices and understand the properties that define this space.

Introduction to Symmetric Matrices

A matrix is symmetric if it is equal to its transpose. Mathematically, an (n times n) matrix (A) is symmetric if (A A^T). For a 10 × 10 symmetric matrix, this property means that the entries below the diagonal are the exact mirror images of the entries above the diagonal.

Determining the Dimension of the Vector Space

The dimension of the vector space of (n times n) symmetric matrices can be determined by analyzing the maximum number of linearly independent matrices that can form a basis for this vector space.

Counting the Independent Entries

For a 10 × 10 symmetric matrix, the diagonal elements are symmetric with themselves, and the elements above the diagonal determine the elements below. There are 10 diagonal entries, and the off-diagonal elements can be divided into two equal halves. Therefore, there are 10 45 55 independent entries.

Formula for Dimension

The dimension of the vector space of (n times n) symmetric matrices can be given by the formula (n^2 - n / 2). For a 10 × 10 matrix, the dimension is (10^2 - 10 / 2 100 - 5 55).

Illustrative Example

Consider a 10 × 10 symmetric matrix A of the form:

[begin{bmatrix} a_{11} a_{12} a_{13} cdots a_{110} [4pt] a_{21} a_{22} a_{23} cdots a_{210} [4pt] a_{31} a_{32} a_{33} cdots a_{310} [4pt] vdots vdots vdots ddots vdots [4pt] a_{101} a_{102} a_{103} cdots a_{1010}end{bmatrix}]

Here, (a_{11}, a_{22}, ldots, a_{1010}) are the diagonal elements, and (a_{ij} a_{ji}) for (i eq j).

Conclusion

The dimension of the vector space of 10 × 10 symmetric matrices is 55. This is derived by counting the 10 independent diagonal entries and the 45 independent off-diagonal entries, as each off-diagonal entry is mirrored by its symmetric counterpart below the diagonal.

Key Takeaways

The dimension of the vector space of (n times n) symmetric matrices is given by (n^2 - n / 2). For a 10 × 10 symmetric matrix, the dimension is 55. The entries above the diagonal determine the entries below the diagonal due to symmetry.

Keywords: dimension of vector space, symmetric matrices, matrix properties