Symmetric Matrix

Tags: #◇CSMATH2 #◇Linear Algebra #◇Computational Mathematics #◇Matrices #◇Matrix Operations #◇Computer Science #◇Vector Spaces #◇Linear Transformations

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◇Overview

Definition: A square matrix A is called symmetric if A⊤=A. (Handout 2, Page 7)

◇Key Concepts

Definition Remarks

◇Details / Theory

Definition: A square matrix A is called symmetric if A⊤=A. (Handout 2, Page 7)
This means aij​=aji​ for all i,j. Example:
A=[1​2​3 2​4​5 3​5​6​]
(Handout 2, Page 7) Remarks: (Handout 2, Page 7)
If A is symmetric, it must be a square matrix (since A and A⊤ must have the same dimensions for equality).

◇Common Pitfalls

A symmetric matrix must be square ($A^T$ and $A$ must have the same dimensions). $A^T A$ and $AA^T$ are always symmetric for any matrix $A$.

◇Summary

TL;DR — Symmetric: $A^T = A$, meaning $a_{ij} = a_{ji}$. Must be square. Common in applications (covariance matrices, distance matrices). Eigenvalues are always real.

◇Links

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◇References & Navigation

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• Related Notes: ◇Special Types of Square Matrices, ◇Nonsingular Matrix, ◇Matrices Detailed