Question

Find the eigenvalues of the given matrix. For each eigenvalue, give an eigenvector. $$\left[\begin{array}{cc}-3 & 1 \\ 0 & -1\end{array}\right]$$

Short answer

Eigenvalues: \(\lambda_1 = -3\), \(\lambda_2 = -1\). Eigenvectors: \(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\), \(\begin{bmatrix} 1 \\ 2 \end{bmatrix}\).

Step-by-step solution

Define Eigenvalue Equation

The eigenvalues of a matrix \( A \) are found by solving the equation \( \det(A - \lambda I) = 0 \). Here, \( A = \begin{bmatrix} -3 & 1 \ 0 & -1 \end{bmatrix} \) and \( I \) is the identity matrix, \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \). Substitute \( A \) and \( I \) to get \( \begin{bmatrix} -3 - \lambda & 1 \ 0 & -1 - \lambda \end{bmatrix} \).

Calculate Determinant

Determine \( \det(A - \lambda I) \) by calculating the determinant of \( \begin{bmatrix} -3 - \lambda & 1 \ 0 & -1 - \lambda \end{bmatrix} \). The determinant is \((-3-\lambda)(-1-\lambda) - 0 \cdot 1 = (\lambda + 3)(\lambda + 1)\).

Solve for Eigenvalues

Solve \((\lambda + 3)(\lambda + 1) = 0\) for \( \lambda \). This yields two eigenvalues: \( \lambda_1 = -3 \) and \( \lambda_2 = -1 \).

Find Eigenvector for \( \lambda_1 = -3 \)

Substitute \( \lambda_1 = -3 \) into \( A - \lambda I \) to get \( \begin{bmatrix} 0 & 1 \ 0 & 2 \end{bmatrix} \). Solve \( \begin{bmatrix} 0 & 1 \ 0 & 2 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \). From the first equation, \( x_2 = 0 \). Choose \( x_1 = 1 \), giving the eigenvector \( \begin{bmatrix} 1 \ 0 \end{bmatrix} \).

Find Eigenvector for \( \lambda_2 = -1 \)

Substitute \( \lambda_2 = -1 \) into \( A - \lambda I \) to get \( \begin{bmatrix} -2 & 1 \ 0 & 0 \end{bmatrix} \). Solve \( \begin{bmatrix} -2 & 1 \ 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \). From the first equation, \( -2x_1 + x_2 = 0 \) implies \( x_2 = 2x_1 \). Choose \( x_1 = 1 \), giving eigenvector \( \begin{bmatrix} 1 \ 2 \end{bmatrix} \).

Key concepts

Eigenvectors

Eigenvectors are special vectors in matrix algebra that, when a specific linear transformation represented by a matrix is applied to them, result only in a scaling of the vector rather than a change in direction. In other words, if we apply matrix \( A \) to an eigenvector \( \mathbf{v} \), the result is a scalar multiple \( \lambda \mathbf{v} \), where \( \lambda \) is known as an eigenvalue. This relationship can be written as:
\[A\mathbf{v} = \lambda\mathbf{v}\]

• \( \mathbf{v} \) is the eigenvector corresponding to the eigenvalue \( \lambda \).
• Eigenvectors provide a valuable simplified understanding of many linear transformations.

To find eigenvectors, you solve the equation \( (A - \lambda I)\mathbf{v} = \mathbf{0} \), where \( I \) is the identity matrix. This involves determining the null space of the matrix \( A - \lambda I \).

Determinant of a Matrix

The determinant of a matrix is a special value that can be calculated from the elements of a square matrix. It offers insight into several properties of the matrix:

• If the determinant is zero, the matrix is singular, meaning it does not have an inverse.
• The determinant helps in solving systems of linear equations and in calculating eigenvalues.

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \). In the context of finding eigenvalues, we use the determinant to solve \( \det(A - \lambda I) = 0 \). This step yields the eigenvalues of the matrix.
The determinant offers a snapshot of the transformation properties, such as scale changes and rotations, that the matrix represents.

Matrix Algebra

Matrix algebra involves operations that include addition, subtraction, multiplication, and finding determinants and inverses of matrices. Understanding how these operations work is key for solving equations involving matrices.
Matrix multiplication is not as straightforward as multiplication of regular numbers. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second.

• When matrices are multiplied, the element in row \( i \) and column \( j \) of the resulting matrix is the dot product of the \( i \)-th row of the first matrix and the \( j \)-th column of the second matrix.
• This operation is used in the calculation of \( A - \lambda I \) when finding eigenvalues and eigenvectors.

Matrix algebra is foundational to understanding transformations and relationships in multi-dimensional space.

Identity Matrix

The identity matrix is one of the simplest yet most important matrices. It acts as the "1" in matrix algebra, meaning any matrix multiplied by the identity matrix remains unchanged. For a 2x2 matrix, the identity matrix \( I \) is:

• \( I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \)

The identity matrix is always a square matrix with ones on the diagonal and zeros elsewhere. It plays a crucial role in linear algebra, particularly in operations such as solving the eigenvalue equation \( A - \lambda I = 0 \).

• The subtraction of \( \lambda I \) from matrix \( A \) aligns eigenvalues with their respective eigenvectors.
• This subtractive process helps isolate the effect of each eigenvalue on the given transformation.

With its simplicity, the identity matrix is a backbone for creating expressions in matrix equations efficiently.