Problem 32 - Differential Equations and Linear Algebra

Question

Show that an \(n \times n\) symmetric upper triangular matrix is diagonal. [Hint: This amounts to showing that if \(\left.i \neq j, \text { then } a_{i j}=0 .\right]\)

Step-by-Step Solution

Answer

To show that an \(n \times n\) symmetric upper triangular matrix A is diagonal, we use the properties of upper triangular matrices (\(a_{ij} = 0\) for \(i > j\)) and symmetric matrices (\(a_{ij} = a_{ji}\) for all i, j). For any two indices \(i\) and \(j\) with \(i \neq j\), either \(i > j\) or \(i < j\). If \(i > j\), then \(a_{ij} = 0\), and since the matrix is symmetric, this means that \(a_{ji} = a_{ij} = 0\). So, all non-diagonal elements are zero, and therefore, the matrix is diagonal.

Step 1: Name the given matrix

Let's call the given symmetric upper triangular matrix A and its elements as \(a_{ij}\) where i and j both range from 1 to n.

Step 2: Use the properties of the upper triangular matrix

Since A is an upper triangular matrix, we know that if \(i>j\) or \(ij\] Remember that this condition is true only for the entries of the matrix A below the diagonal.

Step 3: Use the properties of the symmetric matrix

Since A is also a symmetric matrix, it satisfies the condition: \[a_{ij} = a_{ji} \quad \text{for all i, j in} \quad 1,2, \cdots ,n\]

Step 4: Create the condition and show that non-diagonal elements are zero

Now, we can use the conditions mentioned in steps 2 and 3 to show that all non-diagonal elements are zero. For any two indices \(i\) and \(j\) with \(i \neq j\), either \(i > j\) or \(i < j\). If \(i > j\), then from step 2, we know that: \[a_{ij} = 0\] However, since the matrix is symmetric, this means that: \[a_{ji} = a_{ij} = 0\] So, all the elements **above** the main diagonal, i.e., the elements with \(i < j\), are also zero. We have now shown that all non-diagonal elements are zero regardless of their position within the matrix. Therefore, an \( n \times n \) symmetric upper triangular matrix is diagonal.

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is a special type of square matrix. In this matrix, all the elements below the main diagonal are zero. This means that only the elements on and above the diagonal can possibly have non-zero values.

• Example of a 3x3 upper triangular matrix:

If we write it in general terms, the matrix A with elements \(a_{ij}\) satisfies:
\[a_{ij} = 0 \quad \text{for} \quad i > j\]This condition ensures that there are no contributions to the calculations from below the diagonal. Thus, it often simplifies solving linear equations. Understanding this property is crucial for examining symmetric matrices combined with upper triangular ones.

Diagonal Matrix

A diagonal matrix is another specific type of matrix, where the entries outside the main diagonal are all zero. Unlike upper triangular matrices, diagonal matrices go a step further by having zeros both above and below the diagonal. A diagonal matrix looks like this:

• Example of a 3x3 diagonal matrix:
• \( \begin{bmatrix} d_{11} & 0 & 0 \ 0 & d_{22} & 0 \ 0 & 0 & d_{33} \end{bmatrix} \)

The main diagonal itself can contain any real numbers. When a symmetric upper triangular matrix is shown to be a diagonal matrix, like in the original exercise, it means that all non-diagonal elements are confirmed as zeros due to their symmetric and upper triangular properties, leaving only the diagonal elements non-zero.

Matrix Symmetry

Matrix symmetry is a characteristic where a matrix is identical to its transpose. A symmetric matrix will satisfy the condition \(a_{ij} = a_{ji}\) for all its elements. This means the matrix reflects along its main diagonal.

• For symmetric matrices:
• \(A = A^T\)

In the context of our exercise, the symmetric nature of the given matrix ensured that the zeroing of elements below the diagonal extended logically to elements above it. Thus, symmetry was pivotal in collapsing the non-diagonal elements into zeros, validating the transformation of the matrix into a diagonal form. Upholding symmetry in calculations helps maintain consistency and reveals underlying matrix properties relevant to solving such exercises.