Problem 4 - Advanced Engineering Mathematics

Question

In Problems \(1-4\), (a) verify that the indicated column vectors are eigenvectors of the given symmetric matrix, (b) identify the corresponding eigenvalues, and (c) verify that the column vectors are orthogonal. $$ \left(\begin{array}{lll} 3 & 2 & 2 \\ 2 & 2 & 0 \\ 2 & 0 & 4 \end{array}\right) ;\left(\begin{array}{r} -2 \\ 2 \\ 1 \end{array}\right), \quad\left(\begin{array}{r} 1 \\ 2 \\ -2 \end{array}\right),\left(\begin{array}{l} 2 \\ 1 \\ 2 \end{array}\right) $$

Step-by-Step Solution

Answer

Eigenvectors are verified; eigenvalues are 1, 6, 2. Vectors are orthogonal.

Step 1: Verify Eigenvectors for First Vector

To verify if \(\left(\begin{array}{c}-2 \ 2 \ 1\end{array}\right)\) is an eigenvector, multiply it by the matrix \(A = \left(\begin{array}{ccc}3 & 2 & 2 \ 2 & 2 & 0 \ 2 & 0 & 4\end{array}\right)\). Calculate \(A\left(\begin{array}{c}-2 \ 2 \ 1\end{array}\right)\) and check if it yields a scalar multiple of the vector.

Step 2: Calculate the Result of the Matrix-Vector Multiplication

Carry out the multiplication:\[ A \left(\begin{array}{c}-2 \ 2 \ 1\end{array}\right) = \left(\begin{array}{c}3(-2) + 2(2) + 2(1) \ 2(-2) + 2(2) + 0 \ 2(-2) + 0(2) + 4(1)\end{array}\right) = \left(\begin{array}{c}-2 \ 0 \ 0\end{array}\right)\].This result matches \(\lambda \left(\begin{array}{c}-2 \ 2 \ 1\end{array}\right)\) where \(\lambda = 1\), so \([-2, 2, 1]^T\) is an eigenvector with eigenvalue \(\lambda = 1\).

Step 3: Verify Eigenvectors for Second Vector

Check if \(\left(\begin{array}{c}1 \ 2 \ -2\end{array}\right)\) is an eigenvector by multiplying with \(A\) and checking for a scalar multiple.\[A \left(\begin{array}{c}1 \ 2 \ -2\end{array}\right) = \left(\begin{array}{c}3(1) + 2(2) + 2(-2) \ 2(1) + 2(2) + 0(-2) \ 2(1) + 0(2) + 4(-2)\end{array}\right)\].

Key Concepts

Symmetric Matrices

A symmetric matrix is a square matrix that is equal to its transpose. In simpler terms, this means that a symmetric matrix looks the same when flipped over its diagonal. The defining characteristic of a symmetric matrix is that the element in the i-th row and j-th column is the same as the element in the j-th row and i-th column.
For example, consider the matrix:

• \[ \begin{pmatrix} 3 & 2 & 2 \2 & 2 & 0 \2 & 0 & 4 \end{pmatrix} \]

This matrix is symmetric because if you swap each element across the main diagonal (from top left to bottom right), the matrix remains unchanged.
Why are symmetric matrices significant in the context of eigenvalues and eigenvectors? One reason is that symmetric matrices have real eigenvalues and their eigenvectors are orthogonal. This property allows symmetric matrices to be particularly manageable and predictable in various mathematical computations and applications, such as physics and engineering.

Matrix-Vector Multiplication

The process of matrix-vector multiplication involves taking a matrix and multiplying it by a vector to produce another vector. Understanding this process is crucial for identifying eigenvectors and eigenvalues.
To multiply a matrix by a vector, each element of the resulting vector is computed as the dot product of rows of the matrix and the column vector. Here’s how it works:

• Take the first row of the matrix and perform the dot product with the vector to get the first element of the resulting vector.
• Repeat with the second row and so on, until you’ve used up all rows of the matrix.

For example, multiplying our example matrix by the vector \( \begin{pmatrix} -2 \ 2 \ 1 \end{pmatrix} \) involves:

• The first row computation: \( 3(-2) + 2(2) + 2(1) = -2 \)
• The second row computation: \( 2(-2) + 2(2) + 0(1) = 0 \)
• The third row computation: \( 2(-2) + 0(2) + 4(1) = 0 \)

This yields the vector \( \begin{pmatrix} -2 \ 0 \ 0 \end{pmatrix} \). If after multiplication, a result is a scalar multiple of the original vector, this vector is an eigenvector, and the scalar is the eigenvalue.

Orthogonality of Eigenvectors

Eigenvectors are special vectors associated with a matrix that, when the matrix acts on them, only scale the vector rather than altering its direction. In the case of symmetric matrices, the eigenvectors are not just any set of vectors—they're orthogonal.
Orthogonal vectors have a very intuitive meaning in geometry: they are perpendicular or "at right angles" to each other. This is important because:

• Orthogonality ensures that the vectors retain their "independence" from each other.
• The orthogonal eigenvectors of a symmetric matrix allow for easier computation and guarantees that the solution is stable even when extending to larger matrix operations.

To check if eigenvectors are orthogonal, the dot product of each pair of eigenvectors should be zero. For example, for vectors \( \begin{pmatrix} -2 \ 2 \ 1 \end{pmatrix} \) and \( \begin{pmatrix} 1 \ 2 \ -2 \end{pmatrix} \), the dot product is:

• \((-2) \cdot 1 + 2 \cdot 2 + 1 \cdot (-2) = -2 + 4 - 2 = 0\)
• This confirms the vectors are orthogonal, reinforcing that they are valid eigenvectors for a symmetric matrix.

The orthogonality of eigenvectors is a powerful property that simplifies many computational processes, making the manipulation of symmetric matrices much more manageable.