Upper triangular matrix

An upper triangular matrix is a cornerstone concept in linear algebra, essential for solving systems of equations, matrix decompositions, and eigenvalue problems. 

What is an Upper Triangular Matrix?

Definition:
An upper triangular matrix is a square matrix where all elements below the principal diagonal are zero. For a matrix U=[Uij] of order n×n:

uij=0for alli>j.

Here, ii is the row index, and jj is the column index.

Example of a 3x3 Upper Triangular Matrix:

All elements below the diagonal (positions (2,1),(3,1),(3,2)(2,1),(3,1),(3,2)) are zero.

In Hindi (त्रिकोणीय आव्यूह):
एक ऊपरी त्रिकोणीय आव्यूह वह वर्ग आव्यूह है जिसमें मुख्य विकर्ण के नीचे के सभी अवयव शून्य होते हैं। उदाहरण:

Types of Upper Triangular Matrices

1. Strictly Upper Triangular Matrix:
   • All diagonal and below-diagonal elements are zero.

   • Example:

Key Properties

1. Determinant:
   The determinant is the product of diagonal elements:

Examples of Upper Triangular Matrices

1. 3x3 Upper Triangular Matrix:

Determinant: 1×5×9=451×5×9=45.

Determinant: 2×7×6×4=3362×7×6×4=336.

Applications

1. Solving Linear Systems:
   Used in Gaussian elimination to simplify systems like Ux=bUx=b.
2. Matrix Decompositions:
   Key in LU decomposition, where a matrix is split into lower (L) and upper (U) triangular matrices.
3. Eigenvalue Computations:
   Simplifies eigenvalue problems in algorithms like the QR algorithm.
4. Computer Graphics:
   Represents geometric transformations like scaling and shearing.

How to Convert a Matrix to Upper Triangular Form

Example: Convert A=[123456789]A=147258369:

1. Step 1: Subtract 4×Row 14×Row 1 from Row 2.
2. Step 2: Subtract 7×Row 17×Row 1 from Row 3.
3. Step 3: Subtract 2×Row 22×Row 2 from Row 3.
   Result:

Read Other Math Articles