Upper and Lower Triangular Matrix

Triangular Matrix:

A square matrix in which all the entries upper or below the maim diagonals elements are zero is said to be a triangular matrix. The triangular matrix must be a square matrix that means the triangular matrix has the same number of rows and columns.

There are two types of triangular matrices. the upper triangular matrix and lower triangular matrix

Types of a Triangular Matrix:

Upper Triangular Matrix:

Upper triangular matrix

A square matrix in which entries below the main diagonals are zero is called an upper triangular matrix.

For Example:

\begin{bmatrix}5&3&2&5\\0&7&7&9\\0&0&4&1\\0&0&0&8\end{bmatrix}

Lower Triangular Matrix:

lower triangular matrix

A square matrix in which the entries above the main diagonals are zero is called a lower triangular matrix.

For Examples:

\begin{bmatrix}9&0&0&0\\6&2&0&0\\7&3&8&0\\3&5&9&6\end{bmatrix} \begin{bmatrix}6&0&0\\9&1&0\\7&3&7\end{bmatrix}

Notation

The capital word U is the notation of the upper triangular matrix it is also called the right triangular matrix. And the capital word L is used for the lower triangular matrix it is also called the left triangular matrix.

Properties of an Upper Triangular Matrix:

  • If we added or subtract two upper triangular matrices then the matrix that we obtain is also a upper triangular matrix
  • If we multiply two upper triangular matrix the resultant matrix also upper triangular matrix
  • If we take inverse of upper triangular matrix then the resultant matrix is upper triangular matrix
  • Transpose¬† of upper triangular matrix is lower triangular matrix such that U^t = L.
  • The upper triangular remain unchanged when we multiply it by any scalar quantity .

Properties of a Lower Triangular Matrix:

  • If we added or subtract two lower triangular matrices then the resultant matrix is a lower triangular matrix.
  • If we multiply two lower triangular matrix the resultant matrix also lower triangular matrix
  • If we take inverse of lower triangular matrix then the resultant matrix is lower triangular matrix
  • Transpose¬† of lower triangular matrix is lower triangular matrix such that L^t</span> = U .
  • The lower triangular remain unchanged when we multiply it by any scalar quantity .

Some other types of a triangular matrix

Unit Triangular Matrix:

In any upper or lower triangular matrix if the main diagonals entries are 1 then it is called unit (upper or lower) triangular matrix. Mostly it is called normed triangular matrix

\begin{bmatrix}1&0&0&0\\6&1&0&0\\7&3&1&0\\3&5&9&1\end{bmatrix}\; lower unit triangular matrix

\begin{bmatrix}1&6&7&8\\0&1&7&3\\0&0&1&6\\0&0&0&1\end{bmatrix}\; upper unit triangular matrix

Strictly Triangular Matrix:

In any upper or lower triangular matrix if the elements of the main diagonal are zero then it is called strictly upper or lower triangular matrix

\begin{bmatrix}0&6&7&8\\0&0&7&3\\0&0&0&6\\0&0&0&0\end{bmatrix}\; upper strictly triangular matrix.

\begin{bmatrix}0&0&0&0\\5&0&0&0\\6&1&0&0\\9&7&5&0\end{bmatrix}\; lower strictly triangular matrix.

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