Upper Triangular Matrix

The upper triangular matrix has all the elements below the main diagonal as zero. Also, the matrix which has elements above the main diagonal as zero is called a lower triangular matrix.

Upper Triangular Matrix (U)

\(\begin{bmatrix} * & * & * & * \\ & * & * & *\\ & & *&* \\ 0 & & & * \end{bmatrix}\)

Lower Triangular Matrix (L)

\(\begin{bmatrix} * & & & 0\\ * & * & & \\ * & * & * & \\ * & * & * & * \end{bmatrix}\)

From the above representation, we can see the difference between Upper triangular matrix and a lower triangular matrix. As we have known, what are matrices earlier and how they are helpful for mathematical calculations. These triangular matrices are easier to solve, therefore, are very important in numerical analysis. Let us discuss the definition, properties and some examples for the upper triangular matrix.

Upper Triangular Matrix Definition

A matrix is called an upper triangular matrix if it is represented in the form of;

Um,n = \(\left\{\begin{matrix} a_{{m}_n} , for\, m\leq n\\ 0, for\, m>0 \end{matrix}\right\}\)

Also, written in the form of;

U = \(\begin{bmatrix} a_{11} & a_{12} & a_{13} & ….& a_{1n}\\ 0 & a_{22} & a_{23} & …. & a_{2n} \\ 0 & 0 & a_{33} & …. & a_{3n} \\ . & . & . & …. & . \\ 0 & 0 & 0 & …. & a_{nn} \end{bmatrix}\)

The upper triangular matrix is also called as right triangular matrix whereas the lower triangular matrix is also called a left triangular matrix.

Therefore, a square matrix which has zero entries below the main diagonal, are the upper triangular matrix and a square matrix which has zero entries above the main diagonal of the matrix is considered as lower triangular one.

Apart from these two, there are some special form matrices, such as;

  • Unitriangular Matrix
  • Strictly Triangular Matrix
  • Atomic Triangular Matrix

Properties of Upper Triangular Matrix

  • If we add two upper triangular matrices, it will result in an upper triangular matrix itself.
  • If we multiply two upper triangular, it will result in an upper triangular matrix itself.
  • The inverse of the upper triangular matrix remains upper triangular.
  • The transpose of the upper triangular matrix is a lower triangular matrix, UT = L
  • If we multiply any scalar quantity to an upper triangular matrix, then the matrix still remains as upper triangular.

Examples of Upper Triangular Matrix:

  1. \(\begin{bmatrix} 1 & -1 \\ 0 & 2 \\ \end{bmatrix}\)
  2. \(\begin{bmatrix} 1 & 2 & 4 \\ 0 & 3 & 5 \\ 0 & 0 & 6 \\ \end{bmatrix}\)
  3. \(\begin{bmatrix} 31 & -5 & 14 \\ 0 & 20 & -15 \\ 0 & 0 & 45 \\ \end{bmatrix}\)

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Which of the following shows 114 and 258 on a number line?