Identity Matrix

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

---

◇Overview

Definition: The n×n Scalar Matrix whose main diagonal entries are all equal to 1 is called the n×n identity matrix (or identity matrix of order n),...

◇Key Concepts

Definition

◇Details / Theory

Definition: The n×n ◇Scalar Matrix whose main diagonal entries are all equal to 1 is called the n×n identity matrix (or identity matrix of order n), denoted by In​ or simply I. (Handout 2, Page 5)
aij​=0 for i=j, and aii​=1. Examples: (Handout 2, Page 5)
I2​=[1​0 0​1​],I3​=[1​0​0 0​1​0 0​0​1​],I4​=[1​0​0​0 0​1​0​0 0​0​1​0 0​0​0​1​] Property (Multiplicative Identity): If A is an m×n matrix, then: (Handout 2, Page 5)
AIn​=AandIm​A=A`
Personal Note: The identity matrix acts like the number '1' in scalar multiplication.

◇Common Pitfalls

The identity matrix must be square ($n \times n$) — there is no non-square identity. $AI = A$ only when $I$ has the correct size matching $A$'s column count.

◇Summary

TL;DR — The identity matrix $I_n$ is diagonal with all 1s. It is the multiplicative identity: $AI = IA = A$. It is its own inverse and transpose.

◇Links

Related: - ◇Scalar Matrix Source / Reference:

◇References & Navigation

• Previous Topic: ◇Elementary Row Operation
• Next Topic: ◇Important Remarks
• Related Notes: ◇Upper Triangular Matrix, ◇Lower Triangular Matrix, ◇Matrices Detailed