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Needs:
Matrix Transpose
Symmetric Real Matrices
Matrices
Needed by:
Antisymmetric Matrices
Lower Upper Triangular Factorizations
Normal Matrices
Real Matrix Space
Sparsity Patterns
Links:
Sheet PDF
Graph PDF

Symmetric Matrices

Why

We generalize real symmetric matrices to arbitray sets.

Definition

A square matrix is symmetric (a symmetric matrix) if its values do not depend on the order of the indices.

Notation

Suppose $S$ is a nonempty set and $A \in S^{n \times n}$. Then $A$ is symmetric means

\[ A_{ij} = A_{ji} \quad \text{for all } i, j \in \set{1, \dots , n} \]

A symmetric matrix and its transpose are identical. In symbols, $A = A^\top $.

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