Problem 5 - Advanced Engineering Mathematics

Question

In Problems 1-6, determine which of the indicated column vectors are eigenvectors of the given matrix \(\mathbf{A}\). Give the corresponding eigenvalue. $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & -2 & 2 \\ -2 & 1 & -2 \\ 2 & 2 & 1 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right) $$

Step-by-Step Solution

Answer

\( \mathbf{K}_1 \) is not an eigenvector of \( \mathbf{A} \).

Step 1: Understand Eigenvectors and Eigenvalues

An eigenvector of a matrix \( \mathbf{A} \) is a non-zero vector \( \mathbf{v} \) such that when \( \mathbf{A} \) multiplies \( \mathbf{v} \), the result is a scalar multiple of \( \mathbf{v} \). Mathematically, \( \mathbf{A} \mathbf{v} = \lambda \mathbf{v} \) where \( \lambda \) is an eigenvalue.

Step 2: Calculate \( \mathbf{A} \mathbf{K}_1 \)

To determine if \( \mathbf{K}_1 \) is an eigenvector, calculate \( \mathbf{A} \mathbf{K}_1 \). Here, \( \mathbf{A} \) is multiplied by \( \mathbf{K}_1 \). \[ \mathbf{A}\mathbf{K}_1 = \left(\begin{array}{rrr} 1 & -2 & 2 \ -2 & 1 & -2 \ 2 & 2 & 1 \end{array}\right) \left(\begin{array}{c} 0 \ 1 \ 1 \end{array}\right) \].

Step 3: Perform the Matrix Multiplication

Carry out the matrix multiplication: - First row: \( (1)(0) + (-2)(1) + (2)(1) = 0 \)- Second row: \( (-2)(0) + (1)(1) + (-2)(1) = -1 \)- Third row: \( (2)(0) + (2)(1) + (1)(1) = 3 \)Thus, \( \mathbf{A} \mathbf{K}_1 = \left(\begin{array}{c} 0 \ -1 \ 3 \end{array}\right) \).

Step 4: Check for Eigenvector Condition

In order for \( \mathbf{K}_1 \) to be an eigenvector of \( \mathbf{A} \), the result \( \mathbf{A}\mathbf{K}_1 \) should be equal to \( \lambda \mathbf{K}_1 \) for some scalar \( \lambda \). Since \( \mathbf{A} \mathbf{K}_1 = \left(\begin{array}{c} 0 \ -1 \ 3 \end{array}\right) \) does not equal any scalar multiple of \( \mathbf{K}_1 = \left(\begin{array}{c} 0 \ 1 \ 1 \end{array}\right) \), \( \mathbf{K}_1 \) is not an eigenvector of \( \mathbf{A} \).

Key Concepts

Eigenvalues

Eigenvalues are central to understanding how matrices behave when they act on vectors. If you have a matrix \( \mathbf{A} \) and a non-zero vector \( \mathbf{v} \), then \( \lambda \) is an eigenvalue if \( \mathbf{A}\mathbf{v} = \lambda\mathbf{v} \).

This equation tells us that the transformation represented by \( \mathbf{A} \) stretches or shrinks the vector \( \mathbf{v} \) by a factor \( \lambda \), keeping its direction unchanged. This is a powerful concept because it allows us to simplify the transformation matrix when multiplying with other matrices or vectors.

• For example, if \( \lambda = 1 \), \( \mathbf{v} \) remains unchanged in magnitude but retains its direction.
• If \( \lambda = 0 \), the vector collapses to the zero vector.

Eigenvalues can be found by solving the characteristic equation of a matrix, which is derived from its determinant. Understanding eigenvalues helps in studying systems of differential equations, vibrations, and stability in engineering, physics, and economics.

Matrix Multiplication

Matrix multiplication is a way to combine two matrices in a manner that reflects the composition of linear transformations. To multiply a matrix \( \mathbf{A} \) by a vector \( \mathbf{v} \), you align each row of \( \mathbf{A} \) with \( \mathbf{v} \) and perform element-wise multiplication followed by summation.

Here’s how you would carry out the multiplication of \( \mathbf{A} \) and a vector \( \mathbf{K}_1 \):

• First element: Multiply corresponding elements of the first row of \( \mathbf{A} \) with \( \mathbf{K}_1 \) and sum them up.
• Second element: Repeat this for the second row, multiplying with \( \mathbf{K}_1 \) and summing up.
• Third element: Finally, do the same procedure for the third row with \( \mathbf{K}_1 \).

Matrix multiplication is not just limited to vectors; any two matrices can be multiplied, given they have appropriate dimensions. This operation is fundamental in solving linear equations, transforming geometric spaces, and computer graphics applications.

Linear Algebra

Linear algebra forms the backbone of vector spaces and matrix operations. It involves studying vectors, vector spaces, linear transformations, and systems of linear equations. The solutions and methods you work with are encoded in matrices, which are grid-like structures filled with numbers organized in rows and columns.

One of the major goals of linear algebra is to solve linear systems – often represented as \( \mathbf{Ax} = \mathbf{b} \). Here, \( \mathbf{A} \) is a matrix containing coefficients, \( \mathbf{x} \) is a column vector of variables, and \( \mathbf{b} \) is a column vector of constants. Linear algebra employs concepts like vectors spaces, eigenvectors, eigenvalues, and transformations to tackle these equations.

• Vector Spaces: These are collections of vectors that can be scaled and added together.
• Linear Transformations: Functions that map vectors from one space to another, preserving the operations of vector addition and scalar multiplication.
• Applications: Used in computational sciences, optimization problems, structural engineering, and much more.

Understanding linear algebra is crucial because it forms the foundation for more advanced topics in mathematics and is widely used in various fields including machine learning, statistics, and physics.