Problem 33 - 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. Is the product of two \(n \times n\) upper triangular matrices always upper triangular?

Step-by-Step Solution

Answer

Yes, the product of two \(n \times n\) upper triangular matrices is always upper triangular.

Step 1: Define Upper Triangular Matrix

An upper triangular matrix is a square matrix where all the entries below the main diagonal are zero. This means that for a matrix \(A = [a_{ij}]\), if \(i > j\), then \(a_{ij} = 0\).

Step 2: Consider Two Upper Triangular Matrices

Suppose we have two upper triangular matrices, \(A\) and \(B\), both of size \(n \times n\). The product \(C = AB\) will result in another \(n \times n\) matrix.

Step 3: Determine the Product Entries

The entry \(c_{ij}\) of the matrix \(C\) is computed as \(c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}\). To show that \(C\) is upper triangular, we need \(c_{ij} = 0\) for all \(i > j\).

Step 4: Examine Product Structure

For \(i > j\), each term \(a_{ik} b_{kj}\) involves either \(a_{ik} = 0\) (since \(A\) is upper triangular and \(k < i\)) or \(b_{kj} = 0\) (since \(B\) is upper triangular and \(k > j\)). Thus, for \(i > j\), every term in \(c_{ij}\) is zero. This implies that \(C\) is indeed upper triangular.

Step 5: Conclusion

We conclude that the product of two \(n \times n\) upper triangular matrices is always an upper triangular matrix. The diagonal and above-diagonal elements result from non-zero terms, while entries below the diagonal remain zero.

Key Concepts

Understanding Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where two matrices are multiplied to produce a new matrix. The resulting matrix is formed by computing the dot product of rows from the first matrix with columns of the second matrix.

To perform matrix multiplication:

• Ensure that the number of columns in the first matrix equals the number of rows in the second matrix. This is essential for the operation to be valid.
• Each element of the resulting matrix is computed as the sum of the products of corresponding elements from the rows of the first matrix and the columns of the second matrix. This process is repeated for every element in the resulting matrix.
• A common notation for an element in the product matrix is: \[ c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj} \]This formula ensures that the correct elements are multiplied and summed.

In the context of triangular matrices, the structure of zeros simplifies this multiplication, particularly when working with upper or lower triangular matrices. These zeros reduce the number of computations needed.

Decoding Lower Triangular Matrices

A lower triangular matrix is a type of square matrix where every element above the main diagonal is zero.

This means:

• Only the elements on the diagonal or below it can be non-zero.
• They are efficiently represented in computations, where many multiplication operations simplify due to zeros.
• In a matrix notation for an element at position \(a_{ij}\), it holds that \(a_{ij} = 0\) if \(i < j\).

Lower triangular matrices are particularly useful in solving linear equations and matrix factorization, as they allow for straightforward back-substitution methods. This simplicity offers computational advantages, especially with larger datasets, where reducing operations directly saves time and resources.

Exploring Square Matrices

Square matrices are matrices with the same number of rows and columns. These matrices form the foundation for various operations in linear algebra, including determining triangular matrices.

Important aspects of square matrices include:

• They can be of any size \(n \times n\), where \(n\) is any positive integer (denoting the number of rows and columns).
• The main diagonal of a square matrix is the set of elements extending from the top left to the bottom right. Only elements on or directly adjacent to this diagonal are relevant for triangular matrices.
• Square matrices are pivotal in defining transformations and linear mappings, as seen in systems of linear equations, determinants, and eigenvalues.

Understanding square matrices is essential for grasping more complex concepts like matrix decomposition and vector spaces. Their symmetric nature also facilitates unique properties like computing powers of a matrix and checking for invertibility.