Question

A matrix \(A\) and one of its eigenvectors are given. Find the eigenvalue of \(A\) for the given eigenvector. $$ \begin{array}{l} A=\left[\begin{array}{cc} 19 & -6 \\ 48 & -15 \end{array}\right] \\ \vec{x}=\left[\begin{array}{l} 1 \\ 3 \end{array}\right] \end{array} $$

Short answer

The eigenvalue of \( A \) for eigenvector \( \vec{x} \) is 1.

Step-by-step solution

Understanding the Problem

We are given a matrix \( A \) and an eigenvector \( \vec{x} \) associated with \( A \). Our task is to find the eigenvalue \( \lambda \) corresponding to this eigenvector.

Applying the Eigenvalue Equation

The fundamental equation for an eigenvalue \( \lambda \) and an eigenvector \( \vec{x} \) is \( A \vec{x} = \lambda \vec{x} \). By using this equation, we can solve for \( \lambda \).

Computing \( A\vec{x} \)

Calculate the product \( A\vec{x} \) by multiplying the matrix \( A \) by the vector \( \vec{x} \). The calculation is:\[A \vec{x} = \begin{bmatrix} 19 & -6 \ 48 & -15 \end{bmatrix} \begin{bmatrix} 1 \ 3 \end{bmatrix} = \begin{bmatrix} 19 \times 1 + (-6) \times 3 \ 48 \times 1 + (-15) \times 3 \end{bmatrix} = \begin{bmatrix} 19 - 18 \ 48 - 45 \end{bmatrix} = \begin{bmatrix} 1 \ 3 \end{bmatrix}\].

Comparing Results to Find the Eigenvalue

Observe that \( A\vec{x} = \begin{bmatrix} 1 \ 3 \end{bmatrix} \), which is identical to the vector \( \vec{x} \) itself. This means that placing \( \vec{x} \) back into the equation \( A \vec{x} = \lambda \vec{x} \), the only way it matches with \( \begin{bmatrix} 1 \ 3 \end{bmatrix} \) is if \( \lambda = 1 \).

Concluding the Solution

Therefore, the eigenvalue \( \lambda \) for the eigenvector \( \vec{x} \) is \( 1 \). This concludes our solution to finding the eigenvalue for the given eigenvector with matrix \( A \).

Key concepts

Eigenvectors

Eigenvectors are special vectors in the field of linear algebra that, when transformed by a matrix, only change in scale, not in direction. This means they are the vectors for which only the length is altered but not the orientation when you apply a linear transformation represented by a matrix.

• Formally, if you have a matrix \(A\) and a vector \(\vec{x}\), then \(\vec{x}\) is an eigenvector of \(A\) if \(A\vec{x} = \lambda \vec{x}\).
• Here, \(\lambda\) is known as the eigenvalue—the factor by which the eigenvector is scaled.

Eigenvectors do not arise from arbitrary vector transformations but have specific conditions, meaning not all vectors qualify.
Eigenvectors provide insights into the structure of a matrix, revealing symmetries, rotational properties, and are critical in solving systems of differential equations in physics and engineering.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that combines two matrices to produce another matrix. It is essential for transforming vectors and is used in calculating eigenvalues and eigenvectors.
Matrix multiplication involves taking the dot product of rows in the first matrix with columns in the second matrix. For instance, if you have matrix \(A\) and vector \(\vec{x}\) as in our example, multiplying them proceeds as follows:

• For row 1 of \(A\), multiply each element by the corresponding element of \(\vec{x}\) and add the products.
• For row 2 of \(A\), perform the same operation.

This results in another vector where each element is the sum of these products.
Matrix multiplication is not commutative, meaning \(A\vec{x} eq \vec{x}A\) unless special conditions apply. Understanding matrix multiplication is crucial for grasping more advanced mathematical concepts such as linear transformations.

Linear Transformation

A linear transformation is a mapping between two vector spaces that preserves vector addition and scalar multiplication. Matrices, like the one in the original exercise, are representations of these transformations.

• This concept explains how vectors change when applied to matrices. For example, applying matrix \(A\) to vector \(\vec{x}\) transforms \(\vec{x}\) into a new vector.
• Linear transformations can rotate, reflect, stretch, or compress vectors while maintaining the structural properties of vector spaces.

When an eigenvector \(\vec{x}\) is transformed by \(A\), the result is a scaled version of \(\vec{x}\). This is a key feature of eigenvectors in linear transformations, illustrating the predictable nature of how certain vectors behave under such operations.
Linear transformations are core to understanding many mathematical and scientific applications, especially in areas that involve complex systems and multidimensional spaces.