Problem 24 - Differential Equations and Linear Algebra

Question

Give an example of a matrix of the specified form. (In some cases, many examples may be possible.) \(3 \times 3\) upper triangular symmetric matrix.

Step-by-Step Solution

Answer

An example of a \(3 \times 3\) upper triangular symmetric matrix with the specified form is: \[ A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 6 \end{pmatrix} \]

Step 1: Define a generic upper triangular matrix

Let's first define a generic 3x3 upper triangular matrix: \[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & 0 & a_{33} \end{pmatrix} \]

Step 2: Transpose the matrix and set it equal to the original matrix

In order for the matrix to be symmetric, it must be equal to its transpose. Let's write the transpose of the matrix A: \[ A^T = \begin{pmatrix} a_{11} & 0 & 0 \\ a_{12} & a_{22}& 0 \\ a_{13} & a_{23} & a_{33} \end{pmatrix} \] Now, set \(A = A^T\): \[ \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & 0 & a_{33} \end{pmatrix} = \begin{pmatrix} a_{11} & 0 & 0 \\ a_{12} & a_{22}& 0 \\ a_{13} & a_{23} & a_{33} \end{pmatrix} \]

Step 3: Identify the conditions for the matrix to be symmetric

In order for the two matrices to be equal, all of their corresponding elements must be equal. In this case, since the matrix is upper triangular, the elements below the main diagonal are already equal (all zeros). Now we just need to ensure that the off-diagonal elements are equal: 1. \(a_{12} = 0\) 2. \(a_{13} = 0\) 3. \(a_{23} = 0\)

Step 4: Write the symmetric upper triangular matrix

Considering the conditions found in step 3, we can write the symmetric upper triangular Matrix: \[ A = \begin{pmatrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{pmatrix} \]

Step 5: Provide an example

An example of a \(3 \times 3\) upper triangular symmetric matrix with specified form is: \[ A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 6 \end{pmatrix} \]

Key Concepts

Matrix Transpose

In the world of matrices, transposing a matrix is an essential operation. To transpose a matrix, you swap its rows and columns. For a matrix denoted as \(A\), the transpose is represented as \(A^T\).

This operation is straightforward: the element in the ith row and jth column of the original matrix moves to the jth row and ith column of the transpose. For example, if \(A\) is a 3x3 matrix, such as:

• \(A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{pmatrix}\)

Then its transpose, \(A^T\), becomes:

• \(A^T = \begin{pmatrix} a_{11} & a_{21} & a_{31} \ a_{12} & a_{22} & a_{32} \ a_{13} & a_{23} & a_{33} \end{pmatrix}\)

Transpose operations help in various computations and also in figuring out properties like symmetry.

Matrix Symmetry

Matrix symmetry is a fascinating property in matrices. A matrix is considered symmetric if it is identical to its transpose. That means every element \(a_{ij}\) in the matrix is equal to \(a_{ji}\).

This property is essential in many fields, such as physics and computer science, because symmetric matrices often have real eigenvalues and orthogonal eigenvectors.

For a 3x3 matrix to be symmetric, it looks like this:

• \(A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \ a_{12} & a_{22} & a_{23} \ a_{13} & a_{23} & a_{33} \end{pmatrix}\)

Note that the off-diagonal elements mirror each other across the main diagonal. This mirroring forms the basis of matrix symmetry, making it a simple yet powerful concept.

3x3 Matrices

The 3x3 matrices are small yet immensely significant in linear algebra. They consist of three rows and three columns, making them well-suited for illustrating fundamental concepts like matrix operations and transformations.

When dealing with 3x3 matrices, you can perform operations such as addition, subtraction, scalar multiplication, and multiplication with other matrices.

• Each operation has its own set of rules and applications, often useful in spaces like geometry, systems of equations, and even computer graphics.
• A particular case of a 3x3 matrix is the identity matrix: \(I = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}\).
• The identity matrix is like the number 1 in matrix algebra, meaning that multiplying any matrix by the identity returns the original matrix.

The versatility of 3x3 matrices makes them a focal point for introducing matrix theory and applications.