Problem 34 - 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 \(n \times n\) lower triangular matrices always lower triangular?

Step-by-Step Solution

Answer

Yes, the product of two lower triangular matrices is always lower triangular.

Step 1: Understand Lower Triangular Matrices

A lower triangular matrix is a square matrix where all the entries above the main diagonal are zero. For example, if it is an \( n \times n \) matrix, in any row index \( i \), all elements in columns with indices \( j < i \) are non-zero, while all elements with \( j > i \) are zero.

Step 2: Matrix Multiplication Basics

To multiply two matrices, say \( A \) and \( B \), the element in row \( i \) and column \( j \) of the resulting matrix \( C \) is given by the dot product of the \( i \)-th row of \( A \) and the \( j \)-th column of \( B \). This means you sum the products of corresponding elements.

Step 3: Multiply Two Lower Triangular Matrices

Consider two lower triangular matrices \( A \) and \( B \). The entry \( C_{ij} \) of the product \( C = A \times B \) is \( a_{i1}b_{1j} + a_{i2}b_{2j} + \, \ldots \, + a_{in}b_{nj} \). If \( j > i \), then for all terms \( a_{ik}b_{kj} \), either \( a_{ik} \) or \( b_{kj} \) is zero because one of them would be above the diagonal, hence \( C_{ij} = 0 \).

Step 4: Conclusion

Since for \( j > i \), all entries \( C_{ij} \) are zero after the multiplication of two lower triangular matrices, the resulting matrix \( C \) is also a lower triangular matrix.

Key Concepts

Lower Triangular Matrix

A lower triangular matrix is a type of square matrix. It features all elements above its main diagonal set to zero. What does the main diagonal mean? Imagine a line starting from the top-left corner to the bottom-right corner, stretching across the matrix. All numbers above this line in a lower triangular matrix are zeros. For example, in a 3x3 matrix, only the entries on the diagonal and below it (to the left) can contain non-zero values. Anything to the top-right of this diagonal must be zero in this specific matrix type. Lower triangular matrices are quite important in mathematical studies because they simplify solving linear equations, especially when it comes to matrix decompositions. They're an integral part of many algorithms.

Upper Triangular Matrix

An upper triangular matrix is the reverse of its lower triangular counterpart. This means all elements below the main diagonal are zero. Just like a lower triangular matrix, an upper triangular matrix still defines a square matrix, but this time features only zeros under the diagonal. For an example, look at the matrix:

• All entries on and above the diagonal may have any values
• Entries below the diagonal must be zero

These matrices are significant in math as they make certain calculations, like determinants and matrix invertibility, more straightforward. This simplifies many computational tasks that involve matrices.

Square Matrix

A square matrix is a fundamental concept in linear algebra. It refers to a matrix with the same number of rows and columns. Square matrices have unique properties and are essential in various mathematical operations, including matrix multiplication, finding determinants, and eigenvalues. They're like a matrix's basic building block, paving the way for more complex structures like triangular matrices.

Matrix Diagonal

The matrix diagonal serves as a foundation in understanding various types of matrices, such as lower and upper triangular ones. The diagonal is the imaginary line of entries stretching from the top-left corner to the bottom-right corner of a square matrix. Elements on this diagonal can be non-zero or zero, depending on specific matrix arrangements. For instance, in a diagonal matrix, only elements along this diagonal are non-zero. Understanding this concept is crucial since many matrix-related topics, like matrix inverse and eigenvalues, depend heavily on diagonal properties.

Dot Product

The dot product is essential in matrix multiplication. It's the key mechanism for computing the outcome of multiplying two matrices. When you multiply two matrices, the element on row \( i \) and column \( j \) of the new matrix is found by taking the dot product of row \( i \) from the first matrix and column \( j \) from the second. This is done by multiplying each corresponding pair of elements in the row and column, then summing these products. Understanding the dot product aids in comprehending how matrices interact and transform data in various fields of mathematics and engineering.