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Chapter 4: Problem 1

Find the eigenvalues and eigenvectors of the given matrix $A .$$$ A=\left[\begin{array}{rr} 2 & 1 \\ -1 & 2 \end{array}\right] $$

Short Answer

Expert verified
Answer: The eigenvalues of matrix A are 1 and 3. Their respective eigenvectors are $$\begin{bmatrix} -1 \\ 1 \end{bmatrix}$$ and $$\begin{bmatrix} 1 \\ 1 \end{bmatrix}$$.

Step by step solution

01

Find the determinant of (A - 位I)

Form the (A - 位I) matrix, by subtracting 位 from the diagonal elements of the A matrix. $$ (A - \lambda I) = \begin{bmatrix} 2-\lambda & 1 \\ -1 & 2-\lambda \end{bmatrix}. $$ Now, find the determinant of this matrix: $$ \det(A - \lambda I)= (2-\lambda)(2-\lambda) - (-1)(1). $$
02

Solve the equation for 位

Expanding and simplifying the determinant equation, we have $$ \lambda^2 - 4\lambda +3 = 0. $$ Now, we can solve for 位 by factoring the equation: $$ (\lambda - 1)(\lambda - 3) = 0. $$ So, the eigenvalues (位) are 1 and 3.
03

Find the eigenvectors for each eigenvalue

Now, we will find the eigenvectors for each eigenvalue. For 位 = 1, $$ (A - \lambda I)v = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}=0. $$ Since the second row is a multiple of the first, we can ignore it and use the first row as follows: $$ v_1 + v_2 = 0. $$ Choosing v_2 = 1, we have v_1 = -1. Therefore, the eigenvector for 位 = 1 is $$ v_1=\begin{bmatrix} -1 \\ 1 \end{bmatrix}. $$ For 位 = 3, $$ (A - \lambda I)v = \begin{bmatrix} -1 & 1 \\ -1 & -1 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}=0. $$ Again, we can choose one row (the first, in this case) to use in the calculations: $$ -v_1 + v_2 = 0. $$ Choosing v_1 = 1, we have v_2 = 1. Therefore, the eigenvector for 位 = 3 is $$ v_2 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}. $$ In conclusion, the eigenvalues for the given matrix A are 1 and 3, and their respective eigenvectors are $$\begin{bmatrix} -1 \\ 1 \end{bmatrix}$$ and $$\begin{bmatrix} 1 \\ 1 \end{bmatrix}$$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Determinant of a Matrix
Understanding the determinant of a matrix is crucial when diving into linear algebra, particularly when dealing with eigenvalues and eigenvectors. The determinant is a scalar value that provides important information about a square matrix. It can tell us if the matrix has an inverse, what the volume scaling factor of the linear transformation represented by the matrix is, and it plays a key role in characterizing the matrix's eigenvalues.

To calculate the determinant of a 2x2 matrix, such as \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), you would use the formula \( \det(A) = ad - bc \). When finding eigenvalues of a matrix, we apply this calculation to the matrix minus \( \lambda \) times the identity matrix, \( A - \lambda I \), which results in a polynomial that we call the characteristic polynomial.
Characteristic Polynomial
When we find the eigenvalues of a matrix, we encounter the concept of a characteristic polynomial. This polynomial is derived from the determinant of the matrix subtracted by \( \lambda \) times the identity matrix, as seen in the formula \( \det(A - \lambda I) = 0 \). The roots of this polynomial are the eigenvalues of the matrix.

The characteristic polynomial for a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is obtained by computing \( \det(A - \lambda I) \), resulting in \( \lambda^2 - (a+d)\lambda + (ad-bc) = 0 \). For larger matrices, the characteristic polynomial becomes more complex, but the concept remains the same: solve for the values of \( \lambda \) that satisfy the equation, which gives you the matrix's eigenvalues.
Eigenvector Computation
Once we have the eigenvalues from the characteristic polynomial, the next step is to calculate the corresponding eigenvectors. An eigenvector of a matrix is a non-zero vector that changes by only a scalar factor when that matrix is applied to it. This scalar factor is the eigenvalue associated with the eigenvector.

To find an eigenvector, we solve the equation \( (A - \lambda I)v = 0 \) for each eigenvalue \( \lambda \). This is accomplished by plugging each eigenvalue into the matrix equation and reducing it to row-echelon form, which often reveals a system of linear dependencies among the matrix columns.

The resulting free variables allow for multiple solutions鈥攁ny non-zero vector that satisfies the equation is an eigenvector. Typically, we choose values for the free variables to make calculations simpler and to obtain a convenient basis for the eigenspace associated with each eigenvalue.

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Most popular questions from this chapter

Solve the network of Exercise 36 if the source voltage is \(V_{S}(t)=2 e^{-2 t}\) volts.

For the problem in the exercise specified, (a) Write a program that carries out Euler's method. Use a step size of \(h=0.01\). (b) Run your program on the interval given.(c) Check your numerical solution by comparing the first two values, \(\mathbf{y}_{1}\) and \(\mathbf{y}_{2}\), with the hand calculations. (d) Plot the components of the numerical solution on a common graph over the time interval of interest.Exercise 4

The flow system shown in the figure is activated at time \(t=0 .\) Let \(Q_{i}(t)\) denote the amount of solute present in the \(i\) th tank at time \(t\). For simplicity, we assume all the flow rates are a constant \(10 \mathrm{gal} / \mathrm{min}\). It follows that volume of solution in each tank remains constant; we assume the volume to be \(1000 \mathrm{gal}\).The flow system shown in the figure is activated at time \(t=0 .\) Let \(Q_{i}(t)\) denote the amount of solute present in the \(i\) th tank at time \(t\). For simplicity, we assume all the flow rates are a constant \(10 \mathrm{gal} / \mathrm{min}\). It follows that volume of solution in each tank remains constant; we assume the volume to be \(1000 \mathrm{gal}\).(b) Solve the initial value problem defined by the given inflow concentrations and initial conditions. Also, determine \(\lim _{t \rightarrow \infty} \mathbf{Q}(t)\). (c) In Exercises 33 and 34 , the inflow concentrations are constant. Compute the equilibrium solution of the system in part (a). What is the physical significance of this equilibrium solution? (d) In Exercise 35 , the system in part (a) is not autonomous. Graph \(Q_{1}(t)\) and \(Q_{2}(t)\). Determine the maximum amounts of solute in each tank.\(c_{1}=0.5 \mathrm{lb} / \mathrm{gal}, \quad c_{2}=0, \quad Q_{1}(0)=Q_{2}(0)=0\)

Each initial value problem was obtained from an initial value problem for a higher order scalar differential equation. What is the corresponding scalar initial value problem? $$ \mathbf{y}^{\prime}=\left[\begin{array}{rr} 0 & 1 \\ -3 & 2 \end{array}\right] \mathbf{y}+\left[\begin{array}{c} 0 \\ 2 \cos 2 t \end{array}\right], \quad \mathbf{y}(-1)=\left[\begin{array}{l} 1 \\ 4 \end{array}\right] $$

Let \(A=\left[\begin{array}{ll}a & b \\ b & c\end{array}\right]\) be a \((2 \times 2)\) real symmetric matrix. In Exercise 28 of Section \(4.4\), it was shown that such a matrix has only real eigenvalues. We now show that \(A\) has a full set of eigenvectors. Note, by Exercise 30 of Section \(4.4\), that if \(A\) has distinct eigenvalues, then \(A\) has a full set of eigenvectors. Thus, the only case to consider is the case where \(A\) has repeated eigenvalues, \(\lambda_{1}=\lambda_{2}\). (a) If \(\lambda_{1}=\lambda_{2}\), show that \(a=c, b=0\), and therefore \(A=a I\). (b) Exhibit a pair of linearly independent eigenvectors in this case.
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