MIT 18.06SC Lecture 5.2: Transpose

Linear Algebra

MIT 18.06

Matrix Operations

Author

Chao Ma

Published

October 12, 2025

Context

My lecture notes

The transpose operation switches rows to columns. This post covers the transpose portion of Lecture 5.

Definition

Transpose is the operation that switches rows to columns - each \(A_{i,j}\) becomes \(A_{j,i}\)

For a matrix \(A\), the transpose \(A^T\) satisfies: \[ (A^T)_{i,j} = A_{j,i} \]

Symmetric Matrices

A symmetric matrix satisfies \(A = A^T\)

Property: \(RR^T\) is Symmetric

For any matrix \(R\), the product \(RR^T\) is always symmetric.

Proof: \[ (RR^T)^T = (R^T)^T R^T = RR^T \]

Therefore, \(RR^T\) is symmetric.

Source: MIT 18.06SC Linear Algebra, Lecture 5

---
title: "MIT 18.06SC Lecture 5.2: Transpose"
author: "Chao Ma"
date: "2025-10-12"
categories: ["Linear Algebra", "MIT 18.06", "Matrix Operations"]
---

## Context

[My lecture notes](https://github.com/ickma2311/foundations/blob/main/MIT18.06SC/lecture05/5.2_transpose.md)

The transpose operation switches rows to columns. This post covers the transpose portion of Lecture 5.

---

## Definition

**Transpose** is the operation that switches rows to columns - each $A_{i,j}$ becomes $A_{j,i}$

For a matrix $A$, the transpose $A^T$ satisfies:
$$
(A^T)_{i,j} = A_{j,i}
$$

## Symmetric Matrices

A **symmetric matrix** satisfies $A = A^T$

### Property: $RR^T$ is Symmetric

For any matrix $R$, the product $RR^T$ is always symmetric.

**Proof:**
$$
(RR^T)^T = (R^T)^T R^T = RR^T
$$

Therefore, $RR^T$ is symmetric.

---

*Source: MIT 18.06SC Linear Algebra, Lecture 5*