Upper Triangular Matrix
An upper triangular matrix is a square matrix in which all the elements below the main diagonal are zero.
Mathematically, an upper triangular matrix \( U \) of order \( n \times n \) has the form:
\[ U = \begin{bmatrix} u_{11} & u_{12} & u_{13} & \dots & u_{1n} \\ 0 & u_{22} & u_{23} & \dots & u_{2n} \\ 0 & 0 & u_{33} & \dots & u_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & u_{nn} \end{bmatrix} \]
where \( u_{ij} = 0 \) for all \( i > j \).
Properties of Upper Triangular Matrices
1. Determinant of an Upper Triangular Matrix
The determinant of an upper triangular matrix is the product of its diagonal elements:
\[ \det(U) = u_{11} \cdot u_{22} \cdot u_{33} \dots u_{nn} \]
Example
Consider the upper triangular matrix:
\[ U = \begin{bmatrix} 2 & 3 & 1 \\ 0 & 5 & 4 \\ 0 & 0 & 6 \end{bmatrix} \]
The determinant is calculated as:
\[ \det(U) = 2 \times 5 \times 6 = 60 \]
2. Inverse of an Upper Triangular Matrix
The inverse of an upper triangular matrix (if it exists) is also an upper triangular matrix. The matrix is invertible if and only if all diagonal elements are nonzero.
Example
Consider:
\[ U = \begin{bmatrix} 3 & 2 \\ 0 & 4 \end{bmatrix} \]
The inverse is:
\[ U^{-1} = \begin{bmatrix} \frac{1}{3} & -\frac{2}{12} \\ 0 & \frac{1}{4} \end{bmatrix} = \begin{bmatrix} \frac{1}{3} & -\frac{1}{6} \\ 0 & \frac{1}{4} \end{bmatrix} \]
3. Sum of Upper Triangular Matrices
The sum of two upper triangular matrices of the same order is also an upper triangular matrix.
Example
\[ U_1 = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}, U_2 = \begin{bmatrix} 7 & 8 & 9 \\ 0 & 1 & 2 \\ 0 & 0 & 3 \end{bmatrix} \]
\[ U_1 + U_2 = \begin{bmatrix} 8 & 10 & 12 \\ 0 & 5 & 7 \\ 0 & 0 & 9 \end{bmatrix} \]
4. Multiplication of Upper Triangular Matrices
The product of two upper triangular matrices of the same order is also an upper triangular matrix.
Example
\[ U_1 = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}, U_2 = \begin{bmatrix} 4 & 5 \\ 0 & 6 \end{bmatrix} \]
\[ U_1 \times U_2 = \begin{bmatrix} (1 \times 4 + 2 \times 0) & (1 \times 5 + 2 \times 6) \\ (0 \times 4 + 3 \times 0) & (0 \times 5 + 3 \times 6) \end{bmatrix} \]
\[ = \begin{bmatrix} 4 & 17 \\ 0 & 18 \end{bmatrix} \]
5. Eigenvalues of an Upper Triangular Matrix
The eigenvalues of an upper triangular matrix are simply the elements on its main diagonal.
Example
For the matrix:
\[ U = \begin{bmatrix} 7 & 1 & 2 \\ 0 & 5 & 3 \\ 0 & 0 & 9 \end{bmatrix} \]
The eigenvalues are \( 7, 5, 9 \).