Question

Find the eigenvalues and eigenvectors of the following matrices. Do some problems by hand to be sure you understand what the process means. Then check your results by computer. $$\left(\begin{array}{lll}1& 2& 2\\ 2& 3& 0\\ 2& 0& 3\end{array}\right)$$

Short answer

The eigenvalues are $\lambda =-1,\lambda =2,\lambda =7$. Eigenvectors correspondingly can be found by solving $A-\lambda I=0$.

Step-by-step solution

Write the Matrix

Given matrix is $$A=\left(\begin{array}{lllllll}1& 2& 22& 3& 0\text{2}& 0& 3\end{array}\right)$$

Find the Characteristic Polynomial

To find the eigenvalues, calculate the determinant of $A-\lambda I$, where $I$ is the identity matrix and $\lambda$ is a scalar (the eigenvalue).$$det(A-\lambda I)=det\left(\begin{array}{ccccccc}1-\lambda & 2& 22& 3-\lambda & 02& 0& 3-\lambda \end{array}\right)$$

Compute the Determinant

Evaluate the determinant of the matrix from Step 2:$$det\left(\begin{array}{ccccccc}1-\lambda & 2& 22& 3-\lambda & 02& 0& 3-\lambda \end{array}\right)=(1-\lambda )((3-\lambda )(3-\lambda )-(0)(2))-2(2(3-\lambda )-0)+2(2.0-2(3-\lambda ))$$Simplify the expression.

Solve the Characteristic Polynomial

Solve the equation obtained to find the eigenvalues:$$-(\lambda -1)({\lambda }^2-7\lambda +4)$$The roots of this polynomial are the eigenvalues.

Find the Eigenvectors

For each eigenvalue $\lambda$, substitute $\lambda$ back into the equation $A-\lambda I=0$, and solve for the eigenvector $v$.For example, let $\lambda =7$, solve $$A-7I$$ for $v$. Repeat for each eigenvalue.

Check using a Computer

Use a computer tool or calculator to verify the results of the eigenvalues and eigenvectors.

Key concepts

characteristic polynomial

One of the most important steps in finding the eigenvalues of a matrix is computing the characteristic polynomial. In simple terms, it's a polynomial that is derived from the matrix and helps us determine the eigenvalues. First, you subtract $\lambda \text{I}$ (where $I$ is the identity matrix and $\lambda$ is a scalar) from the original matrix $A$. The resulting matrix is then used to compute the determinant.

To illustrate, consider the matrix $$A=\left(\begin{array}{ccccccc}1& 2& 22& 3& 02& 0& 3\end{array}\right)$$. Its characteristic polynomial is determined by $$det(A-\lambda I)$$. This is computed as follows:

\det(\left($$\begin{array}{ccccccc}1-\lambda & 2& 22& 3-\lambda & 02& 0& 3-\lambda \end{array}$$\right))\.
Solving this determinant gives us the characteristic polynomial. After solving the determinant, we get: $$-(\lambda -1)({\lambda }^2-7\lambda +4)$$

This polynomial, when set to zero and solved for $\lambda$, gives us the eigenvalues of the matrix.

linear algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear transformations, and systems of linear equations. It's a fundamental area in mathematics with applications across physics, engineering, computer science, economics, and many other fields.

In the context of eigenvalues and eigenvectors, linear algebra provides us with tools to analyze and simplify matrices. An eigenvector of a matrix $A$ is a non-zero vector that changes at most by a scalar factor when that matrix is applied to it. The corresponding eigenvalue is the scalar factor by which the eigenvector is scaled.

For example, if $A$ is our matrix and $v$ is an eigenvector, then $$A\mathbf{v}=\lambda \mathbf{v}$$
where $\lambda$ is the eigenvalue. Understanding these concepts helps us solve systems of linear equations and assists in various computations in both theoretical and applied contexts.

matrix determinant

The determinant of a matrix is a scalar value that is computed from the elements of a square matrix. This value helps in determining several properties of the matrix, including whether it is invertible (a non-zero determinant) and insights into the volume scaling factor for linear transformations represented by the matrix.

To compute the determinant of a 3x3 matrix, follow these steps:

• Identify a leading element in the first row.
• Calculate the minor determinants for elements in the first row and form a cofactor expansion.

For the given matrix $$A=\left(\begin{array}{ccccccc}1& 2& 22& 3& 02& 0& 3\end{array}\right)$$,
we calculate the determinant of $A-\lambda I$ using similar steps.
After simplifying, the determinant is $$-(\lambda -1)({\lambda }^2-7\lambda +4)$$. This expression is a polynomial, and setting it to zero helps us find the eigenvalues.

computational verification

Once you have manually computed the eigenvalues and eigenvectors, it's always a good practice to verify your results using computational tools. These tools can save time and help in confirming the accuracy of your calculations.

You can use software like MATLAB, Python (with libraries like NumPy), or online calculators to find the eigenvalues and eigenvectors of matrices. For the given matrix $$A=\left(\begin{array}{ccccccc}1& 2& 22& 3& 02& 0& 3\end{array}\right)$$, inputting this matrix into such tools will yield the eigenvalues and corresponding eigenvectors.
Computational tools follow the same mathematical principles, so understanding the manual process helps in interpreting the results and ensuring correctness. Always cross-check the results to make sure there are no input errors or misinterpretations.