PROJECTENGRAM

SKEW-SYMMETRIC MATRIX

Skew-Symmetric Matrix

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

◇Overview

Definition: A square matrix A is called skew-symmetric (or antisymmetric) if A⊤=−A. (Handout 2, Page 7)

◇Key Concepts

Definition Consequence for Diagonal Entries Remarks

◇Details / Theory

◇Antisymmetric Matrix

Definition: A square matrix A is called skew-symmetric (or antisymmetric) if A⊤=−A. (Handout 2, Page 7)
This means aij​=−aji​ for all i,j. Consequence for Diagonal Entries: If i=j, then aii​=−aii​, which implies 2aii​=0, so aii​=0. All main diagonal entries of a skew-symmetric matrix must be zero. (Handout 2, Page 7 "Remarks") Example:
B=[0​2​3 −2​0​−4 −3​4​0​]
(Handout 2, Page 7) Remarks: (Handout 2, Page 7)
If A is skew-symmetric, it must be a square matrix.

◇Common Pitfalls

All diagonal entries of a skew-symmetric matrix must be zero (aii=−aii⇒aii=0a_{ii} = -a_{ii} \Rightarrow a_{ii} = 0). In real matrices only; complex skew-Hermitian is the generalization.

◇Summary

TL;DR — Skew-symmetric: AT=−AA^T = -A, meaning aij=−ajia_{ij} = -a_{ji} and diagonal entries are zero. Any square matrix can be decomposed into symmetric + skew-symmetric parts.

◇Links

Related: Source / Reference:

◇References & Navigation

CONFIRMATION // 理解度の確認

ACTIVE RECALL VERIFICATION

Select an answer to verify conceptual comprehension and reveal the technical rationale.

Q1

What is the fundamental condition that defines a skew-symmetric (or antisymmetric) matrix AA?

Q2

In terms of individual matrix entries, what does the condition for a skew-symmetric matrix mean?

Q3

What must definitively be true about the main diagonal entries (aiia_{ii}) of a skew-symmetric matrix?