SYMMETRIC MATRIX Meaning and
Definition
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A symmetric matrix is a type of square matrix where the elements are arranged in an order that forms a symmetric pattern along the main diagonal. In other words, it is a square matrix that is equal to its own transpose. The main diagonal of a matrix consists of elements that run from the top left corner to the bottom right corner.
In a symmetric matrix, for any element at the ith row and jth column, the element at the jth row and ith column is also the same. This means that the values are mirrored across the main diagonal. Therefore, the matrix exhibits symmetry with respect to the main diagonal.
A symmetric matrix is often denoted by the capital letter "A" with a subscript "n x n" indicating the order or size of the matrix. It is called an "n x n symmetric matrix" if it has "n" rows and "n" columns. The entries of a symmetric matrix can be real numbers, complex numbers, or even variables.
The symmetry property of a symmetric matrix leads to several important mathematical properties. For example, the main properties of eigenvectors and eigenvalues can be easily derived from symmetric matrices. Additionally, symmetric matrices have useful applications in various fields, such as physics, computer science, and engineering, due to their efficient storage and computational properties.
Etymology of SYMMETRIC MATRIX
The word "symmetric" in "symmetric matrix" is derived from the Latin word "symmetria", which means "agreement in dimensions, form, or proportion". It traces back to the Greek word "synmetria", which has a similar meaning. The prefix "sym-" means "together" or "with", while "metron" means "measure". Therefore, "symmetric" refers to two parts being exactly the same in size, shape, or arrangement. In the context of a matrix, a symmetric matrix is one that is equal to its transpose, possessing symmetry about its main diagonal.
