Problem 35 - Precalculus

Question

Consider the following definitions. A square matrix is said to be an upper triangular matrix if all of its entries below the main diagonal are zero and it is said to be a lower triangular matrix if all of its entries above the main diagonal are zero. For example,$$ E=\left[\begin{array}{rrr} 1 & 2 & 3 \\\ 0 & 4 & -9 \\ 0 & 0 & -5 \end{array}\right] $$ from Exercises 8 - 21 above is an upper triangular matrix whereas $$ F=\left[\begin{array}{ll} 1 & 0 \\\ 3 & 0 \end{array}\right] $$ is a lower triangular matrix. (Zeros are allowed on the main diagonal.) Discuss the following questions with your classmates. Given the matrix $$ A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right] $$ write \(A\) as \(L U\) where \(L\) is a lower triangular matrix and \(U\) is an upper triangular matrix?

Step-by-Step Solution

Answer

The matrix \(A\) can be expressed as \(L = \begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix}\) and \(U = \begin{bmatrix} 1 & 2 \\ 0 & -2 \end{bmatrix}\).

Step 1: Understand the Matrix Decomposition Requirement

We need to express the given matrix \(A\) as a product of a lower triangular matrix \(L\) and an upper triangular matrix \(U\). A lower triangular matrix has all elements above the main diagonal zero, and an upper triangular matrix has all elements below the main diagonal zero.

Step 2: Set Up the Matrices L and U

Let \(L\) be the lower triangular matrix \(\begin{bmatrix} 1 & 0 \ l_{21} & 1 \end{bmatrix}\) and \(U\) be the upper triangular matrix \(\begin{bmatrix} u_{11} & u_{12} \ 0 & u_{22} \end{bmatrix}\). We have placed 1s on the diagonal of \(L\) for simplicity.

Step 3: Perform the Matrix Multiplication LU

Multiply \(L\) and \(U\) to get their product: \[ LU = \begin{bmatrix} 1 & 0 \ l_{21} & 1 \end{bmatrix} \begin{bmatrix} u_{11} & u_{12} \ 0 & u_{22} \end{bmatrix} = \begin{bmatrix} u_{11} & u_{12} \ l_{21}u_{11} & l_{21}u_{12} + u_{22} \end{bmatrix} \].

Step 4: Match the Product Matrix to A

Set the resulting matrix from \(LU\) equal to \(A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}\). This gives us the equations: 1. \(u_{11} = 1\)2. \(u_{12} = 2\)3. \(l_{21}u_{11} = 3\)4. \(l_{21}u_{12} + u_{22} = 4\).

Step 5: Solve for the Values

Using equation 3: \(l_{21} \cdot 1 = 3\), we find \(l_{21} = 3\). Now substitute \(u_{11} = 1\), \(u_{12} = 2\), and \(l_{21} = 3\) into equation 4: \(3 \cdot 2 + u_{22} = 4\). Thus \(u_{22} = 4 - 6 = -2\).

Step 6: Write the Matrices L and U

Now, substitute back the values to write:\(L = \begin{bmatrix} 1 & 0 \ 3 & 1 \end{bmatrix}\) and \(U = \begin{bmatrix} 1 & 2 \ 0 & -2 \end{bmatrix}\).

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is a type of square matrix where all the elements below the main diagonal are zero. This means that only the elements on and above the diagonal can be non-zero. Such matrices are important in simplifying complex matrix operations.
Let's consider an example to illustrate this concept better. Take the matrix \( E = \begin{bmatrix} 1 & 2 & 3 \ 0 & 4 & -9 \ 0 & 0 & -5 \end{bmatrix} \). Notice that all the elements below the main diagonal are zeros, making it an upper triangular matrix.
Upper triangular matrices are particularly useful because their structure simplifies many mathematical operations, such as solving linear equations using methods like back substitution. When you're dealing with matrix decomposition, understanding the properties of upper triangular matrices streamlines the process significantly.

Lower Triangular Matrix

A lower triangular matrix is a square matrix with a specific trait: all its elements above the main diagonal are zero. This configuration allows the elements on and below the diagonal to be non-zero.
Consider the matrix \( F = \begin{bmatrix} 1 & 0 \ 3 & 0 \end{bmatrix} \). Here, all elements above the main diagonal are zeros. Such matrices are essential tools in matrix decomposition algorithms.
Lower triangular matrices are helpful in various computational methods, including forward substitution. They help simplify calculations when solving systems of linear equations. Recognizing this form of matrix is crucial when performing LU decomposition, where separating a matrix into its lower and upper triangular components is a common task.

Matrix Multiplication

Matrix multiplication is a core concept in linear algebra, crucial for performing operations like LU decomposition. It involves multiplying two matrices to produce a new matrix. The product of an \( m \times n \) matrix and an \( n \times p \) matrix results in an \( m \times p \) matrix. However, the actual multiplication process is more nuanced and involves the dot product of rows and columns from the participating matrices.
In the context of our decomposition task, consider matrices \( L \) and \( U \), where \( L \) is lower triangular, and \( U \) is upper triangular. Their multiplication, \( LU \), results in a matrix that we equate to the original matrix \( A \). This process requires careful calculations to ensure that the resulting matrix accurately represents the original.

• Each element of the resulting matrix is the sum of the products of elements from the rows of \( L \) and the columns of \( U \).
• Ensure that the number of columns in \( L \) is equal to the number of rows in \( U \) for the multiplication to be valid.

LU Decomposition

LU decomposition, or factorization, is a method where a matrix is expressed as the product of a lower triangular matrix \( L \) and an upper triangular matrix \( U \). This technique simplifies many operations, such as solving systems of linear equations, and makes other matrix operations more efficient.
The procedure involves ensuring that the initial matrix can be reorganized into these two triangular matrices. With our example, the matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) is decomposed into matrices \( L \) and \( U \):

• \( L = \begin{bmatrix} 1 & 0 \ 3 & 1 \end{bmatrix} \), which is a lower triangular matrix.
• \( U = \begin{bmatrix} 1 & 2 \ 0 & -2 \end{bmatrix} \), which is an upper triangular matrix.

During the decomposition, maintaining the integrity of the matrix operations and ensuring correctness of the values in \( L \) and \( U \) is vital. This process is foundational in computational mathematics and is extensively used in algorithm development for numerical solutions.