Question

Find the eigenvalues of the given matrix. For each eigenvalue, give an eigenvector. $$\left[\begin{array}{ccc}5 & 0 & 0 \\ 1 & 1 & 0 \\ -1 & 5 & -2\end{array}\right]$$

Short answer

Eigenvalues: 5, -2, 1. Eigenvectors: \( \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \) respectively.

Step-by-step solution

Define the characteristic equation

To find the eigenvalues of a matrix, we first find the characteristic equation, which is given by \( \det(A - \lambda I) = 0 \). Here, **A** is the matrix, and \( \lambda \) is a scalar. For our matrix \( A = \begin{bmatrix} 5 & 0 & 0 \ 1 & 1 & 0 \ -1 & 5 & -2 \end{bmatrix} \), \( I \) is the identity matrix of the same size, \( \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \). So, \( A - \lambda I \) becomes \( \begin{bmatrix} 5-\lambda & 0 & 0 \ 1 & 1-\lambda & 0 \ -1 & 5 & -2-\lambda \end{bmatrix} \).

Compute the determinant

The determinant of the matrix \( A - \lambda I \) is computed as follows:\[ \det(A - \lambda I) = (5-\lambda)((1-\lambda)(-2-\lambda) - (5 \cdot 0)) + 0 + 0 \]Calculate the inner determinant:\((1-\lambda)(-2-\lambda) = -2 - \lambda + 2\lambda + \lambda^2 = \lambda^2 + \lambda - 2\).Now, the determinant simplifies to:\[ \det(A - \lambda I) = (5-\lambda)(\lambda^2 + \lambda - 2) \].

Set the determinant to zero

Equate the determinant to zero to find the eigenvalues:\[ (5-\lambda)(\lambda^2 + \lambda - 2) = 0 \].\This gives us the roots as \( \lambda = 5 \) and from the other factor, solve \( \lambda^2 + \lambda - 2 = 0 \) using the quadratic formula or factoring. Factoring this equation, we get \( (\lambda + 2)(\lambda - 1) = 0 \), so \( \lambda = -2 \) and \( \lambda = 1 \). Therefore, the eigenvalues are \( \lambda = 5, -2, 1 \).

Find eigenvectors for each eigenvalue

We now find eigenvectors for each eigenvalue by solving \( (A - \lambda I)\mathbf{v} = \mathbf{0} \):- For \( \lambda = 5 \): \[ (A - 5I) = \begin{bmatrix} 0 & 0 & 0 \ 1 & -4 & 0 \ -1 & 5 & -7 \end{bmatrix} \]. Solve the resulting system; an eigenvector is \( \mathbf{v}_1 = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix} \).- For \( \lambda = -2 \): \[ (A + 2I) = \begin{bmatrix} 7 & 0 & 0 \ 1 & 3 & 0 \ -1 & 5 & 0 \end{bmatrix} \]. Solve the system; an eigenvector is \( \mathbf{v}_2 = \begin{bmatrix} 0 \ 0 \ x \end{bmatrix} \), choose \( x = 1 \): \( \mathbf{v}_2 = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix} \).- For \( \lambda = 1 \): \[ (A - I) = \begin{bmatrix} 4 & 0 & 0 \ 1 & 0 & 0 \ -1 & 5 & -3 \end{bmatrix} \]. Solve the system; an eigenvector is \( \mathbf{v}_3 = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix} \).

Validate eigenvectors

Each eigenvector can be substituted back in the equation \( A\mathbf{v} = \lambda\mathbf{v} \) to ensure they satisfy the condition. These solutions are consistent, confirming correctness.

Key concepts

Characteristic Equation

The characteristic equation is a crucial tool in finding the eigenvalues of a matrix. It comes from the equation \( \det(A - \lambda I) = 0 \), where \( A \) is your matrix and \( \lambda \) represents the eigenvalues. This equation essentially transforms the problem into solving a polynomial where the solutions \( \lambda \) are the eigenvalues.
This process begins by subtracting \( \lambda \) times the identity matrix \( I \) from the matrix \( A \). This results in a new matrix \( (A - \lambda I) \). The function of \( A - \lambda I \) is to form a polynomial whose solutions are the needed eigenvalues.

• It provides the values that make the matrix singular, meaning its determinant is zero.
• The degree of the resulting polynomial is typically equal to the size of the matrix. Here, a 3x3 matrix leads to a cubic polynomial.

Finding the eigenvalues helps in understanding many properties of the matrix, such as stability and oscillation in systems modeled by the matrix.

Eigenvectors

Once the eigenvalues are identified, the next target is to find the eigenvectors. These are non-zero vectors \( \mathbf{v} \) that satisfy the equation \( (A - \lambda I)\mathbf{v} = 0 \), where \( \lambda \) is an eigenvalue.
For each eigenvalue, \( \lambda \), there can be one or many eigenvectors, forming a space called the eigenspace. In essence:

• Eigenvectors are directions that are invariant under the transformation by matrix \( A \).
• After determining eigenvalues, insert them back into \( (A - \lambda I)\mathbf{v} = 0 \) to solve for \( \mathbf{v} \).

Finding eigenvectors typically involves solving a system of linear equations, which can be done using techniques such as row reduction.

Determinant

The determinant is a scalar value that can be computed from a square matrix, and it carries important properties about the matrix itself.
In the process of finding eigenvalues, the determinant appears in the characteristic equation \( \det(A - \lambda I) = 0 \). Here the determinant helps by finding the values of \( \lambda \) that make the matrix \( A - \lambda I \) singular.

• A determinant of zero indicates that the matrix is singular and therefore non-invertible.
• This property helps to detect eigenvalues since they are precisely the values of \( \lambda \) that nullify the determinant.

The determinant can also reflect things like area or volume in geometric terms for transformations represented by matrices.

Matrix Algebra

Matrix algebra encompasses the operations that we perform on matrices, such as addition, multiplication, and finding inverses. In the context of eigenvalues and eigenvectors, matrix algebra involves understanding the interactions between different matrices.
One of the primary activities is subtracting \( \lambda I \) from a matrix \( A \), to create \( (A - \lambda I) \). This operation is fundamental to both creating the characteristic equation and solving for eigenvectors.

• Matrix addition and subtraction are performed element-wise, which means each element of the matrix is manipulated independently.
• Matrix multiplication can be complex, but is vital for transformations and solving equations involving matrices.

Matrix algebra forms the backbone of most calculations involving eigenvalues and eigenvectors, ultimately giving us a way to put theoretical concepts into computational practice.