Math Insight

The behavior of the matrix \begin{align*} A=\left[\begin{matrix}2 & 1\\2 & 3\end{matrix}\right] \end{align*} is demonstrated by the following applet. The solid vectors represent the input vectors. The dashed vectors show the output vectors, which are the result of multiplying the matrix $A$ by the input vectors. Move the endpoints of the solid vectors to see what happens to different vectors.

1. Let's use the applet to get a qualitative description of $A$. There are a few properties of interest. First is the question of scaling, which gives us an idea of the change in the length of vectors. There are several possibilities here. None, some, or all vectors may stay the same length. If not all vectors stay the same length, the ones that don't may either all get longer, all get shorter, or some vectors get longer and some vectors get shorter. Which is the case for the matrix $A$? Compare the lengths of the solid and dashed vectors for different choices of the starting vector to determine this. Some vectors stay the same size, and the rest get longer. Some vectors stay the same size and the rest get shorter. All vectors get shorter. Some vectors get longer, some shorter, and some stay the same size. All vectors stay the same size. All vectors get longer.
   

2. Another qualitative property is rotation. Some matrices will rotate vectors in a consistent direction. Other matrices will rotate different vectors different directions. Which is the case for $A$? All vectors get rotated in the same direction. Some vectors get rotated in different directions than others.
   

3. A third property of interest is the existence of directions which are preserved under $A$. Specifically, we are interested in whether or not there are vectors $\vc{v}$ such that $A\vc{v}$ points in either the same direction as or exact opposite direction from $\vc{v}$. We call such vectors eigenvectors. We often write this as $A\vc{v}=\lambda \vc{v}$, where $\lambda$ is some real number. We will learn how to find these without the assistance of an applet later. For now, we want to find directions in which $A$ stretches or flips vectors. For a $2\times 2$ matrix, there can be zero, one, or two such directions. For this purpose, we count opposite directions as the same, so that vectors $(1,-1)$ and $(-1,1)$ are considered to be the same direction. There is one special case where all vectors are eigenvectors, which is when the matrix is of the form $\begin{bmatrix} a & 0 \\ 0 & a \end{bmatrix}$.

   In how many directions does $A$ have eigenvectors?
   Of these, in how many directions does $A$ flip the eigenvectors?
   

   Use the applet to find the directions where $A$ stretches or flips, i.e. find the eigenvectors. Move the solid arrows until the corresponding dashed arrow is a stretched and/or flipped version of the solid arrow. When this occurs, then we know that $A\vc{v}=\lambda \vc{v}$, where $\vc{v}$ is the vector represented by the solid arrow and $\lambda \vc{v}$ is the stretched and/or flipped vector represented by the dashed arrow. If there are no such directions, enter none in both spots. If there is only one such direction, enter the eigenvector in the first spot and none in the second spot. If there are two directions, enter the eigenvectors in either order. Any scalar multiple will be graded as correct. Remember that to count as two different eigenvectors, the two solid arrows must not point in the same or the opposite directions.

   Eigenvectors:
   ,
   

4. If we know the eigenvectors, we can compute the value of $\lambda$ for which $A\vc{v}=\lambda \vc{v}$. This value of $\lambda$ is known as the eigenvalue. Each coordinate of $A\vc{v}$ should be $\lambda$ times the corresponding coordinate of $\vc{v}$. To determine $\lambda$ for each eigenvector, we just need to divide the first coordinate of $A\vc{v}$ by the first coordinate of $\vc{v}$. First, we need to find $A\vc{v}$. Enter the product of $A$ with each of the eigenvectors found above, in the same order. If you entered none instead of an eigenvector, enter none again.
   $A\vc{v_1}=$
   , $A\vc{v_2}=$
   

   Now that we have $A\vc{v_1}$ and $A\vc{v_2}$, we can compute the eigenvalues. Enter the eigenvalues in the same order as their corresponding eigenvectors.
   Eigenvalues:
   
   

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   Hint

   One option is to find the value of $A$ from the top of the question and multiply it by your chosen eigenvectors to determine $A\vc{v}_1$ and $A\vc{v}_2$. Alternatively, you can open the "Matrix-vector multiplication" section under the applet and let the applet compute the matrix-vector multiplication for you.

   Once you have determined the eigenvector $\vc{v}_1$ and the vector $A\vc{v}_1$, you need to find the eigenvalue $\lambda_1$ that satisfies the equation $A\vc{v}_1 = \lambda_1 \vc{v}_1$. This means, if you multiply the first component of the vector $\vc{v}_1$ by the number $\lambda_1$, you should get the first component of the vector $A\vc{v}_1$. Solve that condition for $\lambda_1$ to determine its value. As a double-check, make sure that the second component of the vector $\vc{v}_1$ times $\lambda_1$ is equal to the second component of the vector $A\vc{v}_1$.

   Repeat using the vectors $\vc{v}_2$ and $A\vc{v}_2$ to determine the second eigenvalue $\lambda_2$.

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1. Let's use the applet to get a qualitative description of $A$. There are a few properties of interest. First is the question of scaling, which gives us an idea of the change in the length of vectors. There are several possibilities here. None, some, or all vectors may stay the same length. If not all vectors stay the same length, the ones that don't may either all get longer, all get shorter, or some vectors get longer and some vectors get shorter. Which is the case for the matrix $A$? Compare the lengths of the solid and dashed vectors for different choices of the starting vector to determine this. Some vectors stay the same size, and the rest get longer. All vectors stay the same size. All vectors get shorter. All vectors get longer. Some vectors stay the same size and the rest get shorter. Some vectors get longer, some shorter, and some stay the same size.
   

2. Another qualitative property is rotation. Some matrices will rotate vectors in a consistent direction. Other matrices will rotate different vectors different directions. Which is the case for $A$? All vectors get rotated in the same direction. Some vectors get rotated in different directions than others.
   

3. A third property of interest is the existence of directions which are preserved under $A$. Specifically, we are interested in whether or not there are vectors $\vc{v}$ such that $A\vc{v}$ points in either the same direction as or exact opposite direction from $\vc{v}$. We call such vectors eigenvectors. We often write this as $A\vc{v}=\lambda \vc{v}$, where $\lambda$ is some real number. We will learn how to find these without the assistance of an applet later. For now, we want to find directions in which $A$ stretches or flips vectors. For a $2\times 2$ matrix, there can be zero, one, or two such directions. For this purpose, we count opposite directions as the same, so that vectors $(1,-1)$ and $(-1,1)$ are considered to be the same direction.

   In how many directions does $A$ have eigenvectors?
   Of these, in how many directions does $A$ flip the eigenvectors?
   

   Use the applet to find the directions where $A$ stretches or flips, i.e. find the eigenvectors. Move the solid arrows until the corresponding dashed arrow is a stretched and/or flipped version of the solid arrow. When this occurs, then we know that $A\vc{v}=\lambda \vc{v}$, where $\vc{v}$ is the vector represented by the solid arrow and $\lambda \vc{v}$ is the stretched and/or flipped vector represented by the dashed arrow. If there are no such directions, enter none in both spots. If there is only one such direction, enter the eigenvector in the first spot and none in the second spot. If there are two directions, enter the eigenvectors in either order. Any scalar multiple will be graded as correct. Remember that to count as two different eigenvectors, the two solid arrows must not point in the same or the opposite directions.

   Eigenvectors:
   ,
   

4. If we know the eigenvectors, we can compute the value of $\lambda$ for which $A\vc{v}=\lambda \vc{v}$. This value of $\lambda$ is known as the eigenvalue. Each coordinate of $A\vc{v}$ should be $\lambda$ times the corresponding coordinate of $\vc{v}$. To determine $\lambda$ for each eigenvector, we just need to divide the first coordinate of $A\vc{v}$ by the first coordinate of $\vc{v}$. First, we need to find $A\vc{v}$. Enter the product of $A$ with each of the eigenvectors found above, in the same order. If you entered none instead of an eigenvector, enter none again.
   $A\vc{v_1}=$
   , $A\vc{v_2}=$
   

   Now that we have $A\vc{v_1}$ and $A\vc{v_2}$, we can compute the eigenvalues. Enter the eigenvalues in the same order as their corresponding eigenvectors.
   Eigenvalues:
   
   

1. Let's use the applet to get a qualitative description of $A$. There are a few properties of interest. First is the question of scaling, which gives us an idea of the change in the length of vectors. There are several possibilities here. None, some, or all vectors may stay the same length. If not all vectors stay the same length, the ones that don't may either all get longer, all get shorter, or some vectors get longer and some vectors get shorter. Which is the case for the matrix $A$? Compare the lengths of the solid and dashed vectors for different choices of the starting vector to determine this. All vectors get shorter. All vectors stay the same size. All vectors get longer. Some vectors stay the same size, and the rest get longer. Some vectors stay the same size and the rest get shorter. Some vectors get longer, some shorter, and some stay the same size.
   

2. Another qualitative property is rotation. Some matrices will rotate vectors in a consistent direction. Other matrices will rotate different vectors different directions. Which is the case for $A$? Not all vectors are rotated in the same direction. All vectors get rotated in the same direction.
   

3. A third property of interest is the existence of directions which are preserved under $A$. Specifically, we are interested in whether or not there are vectors $\vc{v}$ such that $A\vc{v}$ points in either the same direction as or exact opposite direction from $\vc{v}$. We call such vectors eigenvectors. We often write this as $A\vc{v}=\lambda \vc{v}$, where $\lambda$ is some real number. We will learn how to find these without the assistance of an applet later. For now, we want to find directions in which $A$ stretches or flips vectors. For a $2\times 2$ matrix, there can be zero, one, or two such directions. For this purpose, we count opposite directions as the same, so that vectors $(1,-1)$ and $(-1,1)$ are considered to be the same direction.

   In how many directions does $A$ have eigenvectors?
   Of these, in how many directions does $A$ flip the eigenvectors?
   

   Use the applet to find the directions where $A$ stretches or flips, i.e. find the eigenvectors. Move the solid arrows until the corresponding dashed arrow is a stretched and/or flipped version of the solid arrow. When this occurs, then we know that $A\vc{v}=\lambda \vc{v}$, where $\vc{v}$ is the vector represented by the solid arrow and $\lambda \vc{v}$ is the stretched and/or flipped vector represented by the dashed arrow. If there are no such directions, enter none in both spots. If there is only one such direction, enter the eigenvector in the first spot and none in the second spot. If there are two directions, enter the eigenvectors in either order. Any scalar multiple will be graded as correct. Remember that to count as two different eigenvectors, the two solid arrows must not point in the same or the opposite directions.

   Eigenvectors:
   ,
   

4. If we know the eigenvectors, we can compute the value of $\lambda$ for which $A\vc{v}=\lambda \vc{v}$. This value of $\lambda$ is known as the eigenvalue. Each coordinate of $A\vc{v}$ should be $\lambda$ times the corresponding coordinate of $\vc{v}$. To determine $\lambda$ for each eigenvector, we just need to divide the first coordinate of $A\vc{v}$ by the first coordinate of $\vc{v}$. First, we need to find $A\vc{v}$. Enter the product of $A$ with each of the eigenvectors found above, in the same order. If you entered none instead of an eigenvector, enter none again.
   $A\vc{v_1}=$
   , $A\vc{v_2}=$
   

   Now that we have $A\vc{v_1}$ and $A\vc{v_2}$, we can compute the eigenvalues. Enter the eigenvalues in the same order as their corresponding eigenvectors.
   Eigenvalues:
   
   

1. Let's use the applet to get a qualitative description of $A$. There are a few properties of interest. First is the question of scaling, which gives us an idea of the change in the length of vectors. There are several possibilities here. None, some, or all vectors may stay the same length. If not all vectors stay the same length, the ones that don't may either all get longer, all get shorter, or some vectors get longer and some vectors get shorter. Which is the case for the matrix $A$? Compare the lengths of the solid and dashed vectors for different choices of the starting vector to determine this. Some vectors stay the same size and the rest get shorter. All vectors stay the same size. All vectors get longer. Some vectors get longer, some shorter, and some stay the same size. Some vectors stay the same size, and the rest get longer. All vectors get shorter.
   

2. Another qualitative property is rotation. Some matrices will rotate vectors in a consistent direction. Other matrices will rotate different vectors different directions. Which is the case for $A$? All vectors get rotated in the same direction. Some vectors get rotated in different directions than others.
   

3. A third property of interest is the existence of directions which are preserved under $A$. Specifically, we are interested in whether or not there are vectors $\vc{v}$ such that $A\vc{v}$ points in either the same direction as or exact opposite direction from $\vc{v}$. We call such vectors eigenvectors. We often write this as $A\vc{v}=\lambda \vc{v}$, where $\lambda$ is some real number. We will learn how to find these without the assistance of an applet later. For now, we want to find directions in which $A$ stretches or flips vectors. For a $2\times 2$ matrix, there can be zero, one, or two such directions. For this purpose, we count opposite directions as the same, so that vectors $(1,-1)$ and $(-1,1)$ are considered to be the same direction.

   In how many directions does $A$ have eigenvectors?
   Of these, in how many directions does $A$ flip the eigenvectors?
   

   Use the applet to find the directions where $A$ stretches or flips, i.e. find the eigenvectors. Move the solid arrows until the corresponding dashed arrow is a stretched and/or flipped version of the solid arrow. When this occurs, then we know that $A\vc{v}=\lambda \vc{v}$, where $\vc{v}$ is the vector represented by the solid arrow and $\lambda \vc{v}$ is the stretched and/or flipped vector represented by the dashed arrow. If there are no such directions, enter none in both spots. If there is only one such direction, enter the eigenvector in the first spot and none in the second spot. If there are two directions, enter the eigenvectors in either order. Any scalar multiple will be graded as correct. Remember that to count as two different eigenvectors, the two solid arrows must not point in the same or the opposite directions.

   Eigenvectors:
   ,
   

4. If we know the eigenvectors, we can compute the value of $\lambda$ for which $A\vc{v}=\lambda \vc{v}$. This value of $\lambda$ is known as the eigenvalue. Each coordinate of $A\vc{v}$ should be $\lambda$ times the corresponding coordinate of $\vc{v}$. To determine $\lambda$ for each eigenvector, we just need to divide the first coordinate of $A\vc{v}$ by the first coordinate of $\vc{v}$. First, we need to find $A\vc{v}$. Enter the product of $A$ with each of the eigenvectors found above, in the same order. If you entered none instead of an eigenvector, enter none again.
   $A\vc{v_1}=$
   , $A\vc{v_2}=$
   

   Now that we have $A\vc{v_1}$ and $A\vc{v_2}$, we can compute the eigenvalues. Enter the eigenvalues in the same order as their corresponding eigenvectors.
   

   Eigenvalues:
   
   

   Read only after answered question (Show)(Hide)

   We have a confession to make

   If we told you that you got this question correct, we actually lied to you. It turns out that the matrix $A$ does have eigenvectors and eigenvalues. It's just that the eigenvalues are complex numbers, and the eigenvectors are also composed of complex numbers. We didn't want to bother you with this detail yet, so we thought it just simple to lie and say there aren't any eigenvectors or eigenvalues.

   We hope you'll forgive us this minor indiscretion.

   (Hide)