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Symmetric Matrices#

Definition: Symmetric Matrix

A square matrix \(M \in F^{n \times n}\) is symmetric if the bilinear form \(f: F^n \times F^n \to F\) defined as

\[f(\boldsymbol{x}, \boldsymbol{y}) = \boldsymbol{x}^{\mathsf{T}} M \boldsymbol{y}\]

for all \(\boldsymbol{x}, \boldsymbol{y} \in F^n\) is symmetric.

Theorem: Symmetry via Transpose

A square matrix \(M \in F^{n \times n}\) is symmetric if and only if it is equal to its own transpose:

\[M = M^{\mathsf{T}}\]
Proof

TODO

Theorem: Symmetry of Inverse

If a symmetric matrix is invertible , then its inverse is also symmetric.

Proof

TODO

Definiteness#

Definition: Definiteness

Let \(F\) be an ordered field, let \(M \in F^{n \times n}\) be symmetric and let \(f: F^n \times F^n \to F\) be the symmetric bilinear form defined as

\[f(\boldsymbol{x}, \boldsymbol{y}) = \boldsymbol{x}^{\mathsf{T}} M \boldsymbol{y}\]

for all \(\boldsymbol{x}, \boldsymbol{y} \in F^n\).

We say that \(M\) is positive definite / positive semi-definite / negative definite / negative semi-definite / indefinite if \(f\) is positive definite / positive semi-definite / negative definite / negative semi-definite / indefinite.