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Chapter 7: Problem 1

Find the eigenvalues and eigenvectors of the following matrices, where \(a\) and \(b\) are real and \(b \neq 0\). $$ A=\left[\begin{array}{rr} 1 & 2 \\ -8 & 11 \end{array}\right], \quad B=\left[\begin{array}{rr} a & b \\ -b & a \end{array}\right] $$

Short Answer

Expert verified
Eigenvalues of A: 9, 3 with eigenvectors (1,4), (1,1).\nEigenvalues of B: a卤bi with eigenvectors (1,i), (1,-i).

Step by step solution

01

Find the Eigenvalues of Matrix A

To find the eigenvalues of matrix \( A \), we need to solve the characteristic equation \( \det(A - \lambda I) = 0 \). Start by substituting \( A - \lambda I \) as follows:\[A - \lambda I = \begin{bmatrix} 1-\lambda & 2 \ -8 & 11-\lambda \end{bmatrix}\]The determinant is given by:\[(1-\lambda)(11-\lambda) - (2)(-8) = 0\]Simplifying, we have:\[\lambda^2 - 12\lambda + 27 = 0\]Solve for \( \lambda \) using the quadratic formula \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):\[\lambda = \frac{12 \pm \sqrt{144 - 108}}{2} = \frac{12 \pm 6}{2}\]Thus, the eigenvalues are \( \lambda_1 = 9 \) and \( \lambda_2 = 3 \).
02

Find the Eigenvectors of Matrix A

To find the eigenvectors, substitute each eigenvalue back into \( A - \lambda I \) and solve \((A - \lambda I)\mathbf{v} = \mathbf{0}\) for each eigenvalue.For \( \lambda_1 = 9 \):\[\begin{bmatrix} 1-9 & 2 \ -8 & 11-9 \end{bmatrix} = \begin{bmatrix} -8 & 2 \ -8 & 2 \end{bmatrix}\]The system becomes \(-8x + 2y = 0\), which simplifies to \(y = 4x \). The eigenvector is \( \begin{bmatrix} 1 \ 4 \end{bmatrix} \).For \( \lambda_2 = 3 \):\[\begin{bmatrix} 1-3 & 2 \ -8 & 11-3 \end{bmatrix} = \begin{bmatrix} -2 & 2 \ -8 & 8 \end{bmatrix}\]The system becomes \(-2x + 2y = 0\), which simplifies to \(y = x \). The eigenvector is \( \begin{bmatrix} 1 \ 1 \end{bmatrix} \).
03

Find the Eigenvalues of Matrix B

For matrix \( B \), start with \( B - \lambda I \):\[B - \lambda I = \begin{bmatrix} a-\lambda & b \ -b & a-\lambda \end{bmatrix}\]Next, compute the determinant:\[(a-\lambda)^2 + b^2 = 0\]This simplifies to the characteristic equation:\[\lambda^2 - 2a\lambda + (a^2 + b^2) = 0\]The roots (eigenvalues) are:\[\lambda = a \pm bi\]
04

Find the Eigenvectors of Matrix B

To find the eigenvectors, substitute \( \lambda = a + bi \) back into \( B - \lambda I \):\[\begin{bmatrix} -bi & b \ -b & -bi \end{bmatrix}\]The system becomes \(-bix + by = 0\) and \(-bx - biy = 0\). Solving these, we can choose \( y = i \) leading to the eigenvector \( \begin{bmatrix} 1 \ i \end{bmatrix} \).For \( \lambda = a - bi \), a similar process yields the eigenvector \( \begin{bmatrix} 1 \ -i \end{bmatrix} \).
05

Verify Calculations

Review all calculations to ensure accuracy, double-check eigenvectors to ensure they're solutions to \((A - \lambda I)\mathbf{v} = \mathbf{0}\) or \((B - \lambda I)\mathbf{v} = \mathbf{0}\), and confirm eigenvalue expressions match determinants set to zero for eigenvalue solutions.

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Key Concepts

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

Matrix Algebra
Matrix algebra is a cornerstone of linear algebra, involving operations such as addition, subtraction, and multiplication of matrices. These operations are useful in solving systems of linear equations.
A matrix is essentially a rectangular array of numbers arranged in rows and columns. These numbers can represent different data based on the context, and matrices are heavily used in computer graphics, economics, and statistics.
  • Matrix Addition: Add corresponding elements of two matrices of the same dimension.
  • Matrix Multiplication: Multiply each element of a row of one matrix by each element of a column of another matrix, then sum the products. Unlike addition, this is not commutative.
  • Determinant: A special number obtained from a square matrix that gives insights into the matrix properties, like invertibility.
The eigenvalues and eigenvectors of a matrix, core elements for many applications like stability analysis and quantum mechanics, fundamentally arise from matrix algebra.
Characteristic Equation
The characteristic equation is key to finding eigenvalues of a matrix. Formulated from the matrix, it allows you to determine how the matrix transforms different vectors.
To derive this equation, you begin by considering a matrix \( A \) and its eigenvalues \( \lambda \). The characteristic equation is formulated as:
\[ \det(A - \lambda I) = 0 \]
where \( I \) is the identity matrix and \( \lambda \) is a scalar.

  • The equation \( A - \lambda I \) forms a new matrix.
  • Finding the determinant of this matrix then sets up the equation to solve for \( \lambda \).
This equation characterizes how a matrix behaves when it stretches or shrinks vectors, offering insights into dynamic systems and stability.
Quadratic Formula
The quadratic formula is a reliable tool for solving quadratic equations, especially in the characteristic equation context when seeking matrix eigenvalues.
A typical quadratic equation is expressed as:
\[ ax^2 + bx + c = 0 \]
The quadratic formula to find the roots (solutions) is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Using this formula, you can find the eigenvalues of a matrix when the determinant of \( A - \lambda I \) results in a quadratic equation.
  • The term \( b^2 - 4ac \) is known as the discriminant.
  • A positive discriminant indicates two distinct real solutions.
  • A zero discriminant provides one real solution.
  • A negative discriminant leads to complex solutions.
Understanding this formula is crucial as it aids in finding eigenvalues from characteristic equations with quadratic formations.
Complex Eigenvalues
When dealing with matrices, it's possible to encounter complex eigenvalues, especially when the characteristic equation has a negative discriminant.
Complex numbers take the form \( a + bi \) where \( i \) is the imaginary unit with the property \( i^2 = -1 \). These are critical in various fields like control theory and oscillatory systems.
For a 2x2 matrix setup, as seen with matrix \( B \):
  • The eigenvalues \( a \pm bi \) arise from solving the characteristic equation \( \lambda^2 - 2a\lambda + (a^2 + b^2) = 0 \).
  • Having complex eigenvalues doesn't make them less valid but rather essential in analyzing systems with oscillations or rotations.
Exploring eigenvectors from complex eigenvalues involves using complex arithmetic, leading to elegant solutions. Such analyses are pivotal in understanding wave functions and quantum systems.

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Most popular questions from this chapter

Determine the eigenvalues of the following matrix (i) by direct calculation, (ii) by showing that \(A^{2}=I\). $$ A=\left[\begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right] $$

It should be noted that the spectrum \(\sigma(x)\) of an element \(x\) of a Banach algebra \(A\) depends on \(A\). In fact, show that if \(B\) is a subalgebra of \(A\), then \(\tilde{\sigma}(x) \supset \sigma(x)\), where \(\tilde{\sigma}(x)\) is the spectrum of \(x\) regarded as an element of \(B\).

Illustrate with a simple example that an \(n\)-rowed square matrix may not have eigenvectors which constitute a basis for \(\mathbf{R}^{n}\) (or \(\left.\mathbf{C}^{\prime \prime}\right)\). For instance, consider $$ A=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right] $$

If \(\left(x_{n}\right)\) and \(\left(y_{n}\right)\) are Cauchy sequences in a normed algebra \(A\), show that \(\left(x_{n} y_{n}\right)\) is Cauchy in \(A\). Moreover, if \(x_{n} \longrightarrow x\) and \(y_{n} \longrightarrow y\), show that \(x_{n} y_{n} \longrightarrow x y\).

The behavior of the various parts of the spectrum under extension of an operator is of practical interest. If \(T\) is a bounded linear operator and \(T_{1}\) is a linear extension of \(T\), show that we have \(\sigma_{p}\left(T_{1}\right) \supset \sigma_{p}(T)\) and for any \(\lambda \in \sigma_{p}(T)\) the eigenspace of \(T\) is contained in the eigenspace of \(T_{1}\).
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