Question

A matrix \(A\) and one of its eigenvectors are given. Find the eigenvalue of \(A\) for the given eigenvector. $$ A=\left[\begin{array}{ccc} -12 & -10 & 0 \\ 15 & 13 & 0 \\ 15 & 18 & -5 \end{array}\right] $$ $$ \vec{x}=\left[\begin{array}{c} -1 \\ 1 \\ 1 \end{array}\right] $$

Short answer

The eigenvalue for the given eigenvector is \(-2\).

Step-by-step solution

Understand the Eigenvalue Equation

To find the eigenvalue \( \lambda \) of matrix \( A \) corresponding to an eigenvector \( \vec{x} \), we use the equation \( A\vec{x} = \lambda \vec{x} \). This equation means that when matrix \( A \) multiplies the vector \( \vec{x} \), the result is the same as scaling \( \vec{x} \) by \( \lambda \).

Calculate \( A\vec{x} \)

Perform the matrix-vector multiplication to find \( A\vec{x} \). Multiply each row of matrix \( A \) by vector \( \vec{x} \):\( A \vec{x} = \left[\begin{array}{ccc} -12 & -10 & 0 \ 15 & 13 & 0 \ 15 & 18 & -5 \end{array}\right] \left[\begin{array}{c} -1 \ 1 \ 1 \end{array}\right] = \left[\begin{array}{c} -12(-1) + (-10)(1) + 0(1) \ 15(-1) + 13(1) + 0(1) \ 15(-1) + 18(1) - 5(1) \end{array}\right].\)

Simplify the Matrix-Vector Product

Simplify the expression calculated in Step 2:- First row: \( -12(-1) + (-10)(1) + 0(1) = 12 - 10 + 0 = 2 \).- Second row: \( 15(-1) + 13(1) + 0(1) = -15 + 13 + 0 = -2 \).- Third row: \( 15(-1) + 18(1) - 5(1) = -15 + 18 - 5 = -2 \).Thus, \( A\vec{x} = \left[\begin{array}{c} 2 \ -2 \ -2 \end{array}\right] \).

Identify the Eigenvalue

The vector \( A\vec{x} = \left[\begin{array}{c} 2 \ -2 \ -2 \end{array}\right] \) must equal \( \lambda \vec{x} = \lambda \left[\begin{array}{c} -1 \ 1 \ 1 \end{array}\right] \). Equating components, we find:1. \( 2 = -\lambda \)2. \( -2 = \lambda \) (from both the second and third components)From the second comparison, we conclude that \( \lambda = -2 \).

Verify Consistency

Check that the eigenvalue \( \lambda = -2 \) satisfies all component equations:- For the first row in \( A\vec{x} \): \( 2 = -(-2) = 2 \).- The second and third rows are already satisfied by \( -2 = \lambda \).Thus, the calculations are consistent and verified. The eigenvalue for the given eigenvector is \(-2\).

Key concepts

Matrix Multiplication

Matrix multiplication is a crucial operation in linear algebra, especially when dealing with eigenvectors and eigenvalues. To multiply a matrix by a vector, you perform the dot product of each row of the matrix with the vector.

Consider the matrix \( A \) and vector \( \vec{x} \). The resulting vector after multiplication has entries that are each obtained by taking the dot product of the vector with each row of the matrix:

• First row calculation: Multiply each corresponding element of the first row of matrix \( A \) by each element of the vector \( \vec{x} \), and sum them up.
• Repeat the process for every subsequent row in matrix \( A \).

Let us illustrate with the given example:
Matrix \( A \) is
\[A = \begin{bmatrix} -12 & -10 & 0 \ 15 & 13 & 0 \ 15 & 18 & -5 \end{bmatrix}\]
Vector \( \vec{x} \) is
\[\vec{x} = \begin{bmatrix} -1 \ 1 \ 1 \end{bmatrix}\]
Performing the calculation for each row:

• For the first row: \( (-12)(-1) + (-10)(1) + (0)(1) = 12 - 10 + 0 = 2 \)
• For the second row: \( 15(-1) + 13(1) + 0(1) = -15 + 13 + 0 = -2 \)
• For the third row: \( 15(-1) + 18(1) - 5(1) = -15 + 18 - 5 = -2 \)

So, you end up with the vector \( A \vec{x} = \begin{bmatrix} 2 \ -2 \ -2 \end{bmatrix} \).

Linear Transformations

Linear transformations are essential tools in understanding matrix operations such as those involving eigenvectors and eigenvalues. A linear transformation takes every vector and transforms it linearly, often represented as \( A\vec{x} \) in terms of matrices.

Every linear transformation can be represented by a matrix, and the action is captured by matrix multiplication. Here's how linear transformations relate to the given exercise:

• Input vector \( \vec{x} \): This eigenvector provides the direction that does not rotate during the transformation.
• Outcome vector \( A\vec{x} \): Represents how the vector \( \vec{x} \) is transformed under the matrix. In our example, it's \( \begin{bmatrix} 2 \ -2 \ -2 \end{bmatrix} \).
• Scaling effect \((\lambda)\): Eigenvalue \( \lambda = -2 \) indicates how \( \vec{x} \) scales during the transformation.

Essentially, eigenvectors merely change in size, given eigenvalues, rather than direction when a matrix transformation is applied.

Matrix Algebra

Matrix algebra encompasses all operations involving matrices, such as addition, subtraction, scalar multiplication, and perhaps most importantly, matrix multiplication. It is extensively used in understanding linear systems and transformations.

With eigenvectors and eigenvalues, matrix algebra helps to:

• Identify special vectors (eigenvectors) which when transformed by \( A \) result in vectors that are simply scaled versions of themselves.
• Find the scalar (eigenvalue \( \lambda \)) that describes how much the eigenvector is stretched or compressed.

Key points in this context:

• Eigenvectors are not nullified by the transformation \( A \) but are instead scaled.
• Matrix equations \( A\vec{x} = \lambda\vec{x} \) solidify the relationship between the matrix and vector transformations.
• The specific eigenvalue corresponds to the factor by which its eigenvector is stretched during multiplication.

In this problem's context, we solve for \( \lambda \) by equating the products derived from matrix algebra processes, ensuring consistent scaling of the eigenvector. This aids in understanding the intrinsic geometric transformations matrices enact on vectors.