Triangular Matrices

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

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◇Overview

Upper Triangular Matrix: A square matrix A=[aij​] is called upper triangular if all entries below the main diagonal are zero, i.e., aij​=0 for i>...

◇Key Concepts

◇Upper Triangular Matrix Example ◇Lower Triangular Matrix Example

◇Details / Theory

◇Upper Triangular Matrix: A square matrix A=[aij​] is called upper triangular if all entries below the main diagonal are zero, i.e., aij​=0 for i>j. (Handout 2, Page 6)
Example: U=[4​0​3 0​5​6 0​0​−1​] (Handout 2, Page 6) ◇Lower Triangular Matrix: A square matrix A=[aij​] is called lower triangular if all entries above the main diagonal are zero, i.e., aij​=0 for i<j. (Handout 2, Page 6)
Example: L=[1​0​0 2​5​0 3​−1​2​] (Handout 2, Page 6) Personal Note:
A diagonal matrix is both upper and lower triangular.
The transpose of an upper triangular matrix is a lower triangular matrix, and vice-versa.
The product of two upper (lower) triangular matrices is an upper (lower) triangular matrix.
The determinant of a triangular matrix is the product of its diagonal entries.

◇Common Pitfalls

A matrix that is both upper and lower triangular is a diagonal matrix. Triangular matrices are only defined for square matrices.

◇Summary

TL;DR — Triangular matrices have zeros on one side of the diagonal. Determinant = product of diagonal entries. Transpose of upper triangular is lower triangular and vice versa.

◇Links

Related: - ◇Upper Triangular Matrix - ◇Lower Triangular Matrix Source / Reference:

◇References & Navigation

• Previous Topic: ◇Transpose of a Matrix
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• Related Notes: ◇Size & Dimension of a Matrix, ◇Elementary Row Operation, ◇Matrices Detailed