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Chapter 5: Problem 30

Use the routines MATCOMP and ALLROOTS in IINTEK or use MATLAB to classify the given matrix as real diagonalizable, complex diagonalizable, or not diagonalizable. $$ \left[\begin{array}{rrr} 18 & 25 & -25 \\ 1 & 6 & -1 \\ 18 & 34 & -25 \end{array}\right] $$

Short Answer

Expert verified
The matrix is complex diagonalizable.

Step by step solution

01

Identify Eigenvalues

To begin, find the eigenvalues of the matrix. In MATLAB, use the 'eig' function: \[ A = \left[ \begin{array}{ccc} 18 & 25 & -25 \ 1 & 6 & -1 \ 18 & 34 & -25 \end{array} \right] \]Type in MATLAB: ```matlablambda = eig(A);```This function returns the eigenvalues of the matrix. We use these to determine if the matrix is diagonalizable.
02

Check for Real Eigenvalues and Algebraic Multiplicity

Examine the eigenvalues obtained from the computation. For the matrix to be real diagonalizable, all eigenvalues should be real and their algebraic multiplicity must equal their geometric multiplicity.
03

Determine Eigenvectors and Geometric Multiplicity

Use MATLAB to compute the eigenvectors and ensure the geometric multiplicity matches the algebraic multiplicity for each eigenvalue. This can be done by typing: ```matlab [V,D] = eig(A); ``` The columns of V are the eigenvectors. Check that there is a independent eigenvector for each distinct eigenvalue.
04

Check Diagonalizability Criteria

If the eigenvalues are real and the geometric multiplicity matches the algebraic multiplicity for each eigenvalue, the matrix is real diagonalizable. If some eigenvalues are complex conjugates but can be paired with their eigenvectors to form a full set of linearly independent vectors, the matrix is complex diagonalizable. If these conditions are not met, it is not diagonalizable.

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Key Concepts

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

Eigenvalues
Eigenvalues are crucial in understanding whether a matrix can be diagonalized. They are essentially the scalars that help us understand the matrix operation. To find eigenvalues of a matrix, you solve the characteristic equation \( \det(A - \lambda I) = 0 \), where \( A \) is your matrix and \( \lambda \) represents the eigenvalues.
By determining these eigenvalues, you begin the process of determining the matrix's behavior when amplified in its own space. Each eigenvalue can provide insights into the scaling factor applied to its associated eigenvector.
Eigenvectors
Eigenvectors are non-zero vectors that only change by a scalar (the eigenvalue) when a matrix is applied to them. Given a matrix \( A \), associated with each eigenvalue \( \lambda \) is an eigenvector \( v \), which satisfies the equation \( Av = \lambda v \).
Finding eigenvectors involves solving for \( v \) in the equation \( (A - \lambda I)v = 0 \). Eigenvectors are pivotal since their span provides directions that remain intact under linear transformations expressed by the matrix.
  • An eigenvector points in a direction where the transformation by \( A \) only scales it by its eigenvalue.
  • Eigenvectors form the basis in the process of diagonalizing matrices by converting them into a diagonal state using similarity transformations.
Geometric Multiplicity
Geometric multiplicity refers to the number of linearly independent eigenvectors corresponding to an eigenvalue. It tells us how many dimensions the eigenspace associated with a particular eigenvalue encompasses.
The geometric multiplicity of an eigenvalue \( \lambda \) is calculated by solving \( (A - \lambda I)v = 0 \) and counting how many independent solutions (eigenvectors) the equation has.
  • It is always equal to or less than the algebraic multiplicity.
  • Geometric multiplicity provides an indication of the potential diagonalizability of a matrix; when it equals the algebraic multiplicity, there's a higher chance that the matrix is diagonalizable.
Algebraic Multiplicity
Algebraic multiplicity is the number of times an eigenvalue appears in the characteristic equation. This indicates how the eigenvalue behaves as a root within the polynomial equation derived from the determinant \( \det(A - \lambda I) = 0 \).
The algebraic multiplicity gives us the repeated count of an eigenvalue which provides insights into the linear characteristics of the eigen transformations as prescribed by the matrix.
  • Algebraic multiplicity is always a positive integer.
  • For a matrix to be diagonalizable, each eigenvalue's algebraic multiplicity should be equal to its geometric multiplicity.
Matching the algebraic and geometric multiplicity indicates that eigenvalues are adequately represented by independent eigenvectors.

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Most popular questions from this chapter

Let \(A\) be an \(n \times n\) matrix, and let \(C\) be an invertible \(n \times n\) matrix. Show that the eigenvalues of \(A\) and of \(C^{-1} A C\) are the same. [Hint: Show that the characteristic polynomials of the two matrices are the same.]

Use the routine MATCOMP in LINTEK, or MATLAB, to illustrate the Cayley- Hamilton theorem for the matrii $$ \left[\begin{array}{rrrr} -2 & 4 & 6 & -1 \\ 5 & -8 & 3 & 2 \\ 11 & -3 & 7 & 1 \\ 0 & -5 & 9 & 10 \end{array}\right] \text {. } $$

The routine ALLROOTS in LINTEK can be used to find both real and complex roots of a polynomial. The program uses Newton's method, which finds a solution by successive approximations of the polynomial function by a linear one. ALLROOTS is designed so that the user can watch the approximations approach a solution. Of course, a program such as MATLAB, which is designed for research, simply spits out the answers. In Exercises 52-55, either 2\. use the command eig(A) in MATLAB to find all eigenvalues of the matrix or b. first use MATCOMP in LINTEK to find the characteristic equation of the given matrix. Copy down the equation, and then use ALLROOTS to find all eigenvaiues of the matrix. $$ \left[\begin{array}{rrrr} 21 & -8 & 0 & 32 \\ -14 & 17 & -6 & 9 \\ 15 & 11 & -13 & 16 \\ -18 & 30 & 43 & 31 \end{array}\right] $$

In Exercises 2-16, find the characteristic polynomial, the real eigenvalues, and the corresponding eigenvectors of the given matrix. $$ \left[\begin{array}{rrr} 1 & 0 & 0 \\ -8 & 4 & -5 \\ 8 & 0 & 9 \end{array}\right] $$

Determine whether the given matrix is diagonalizable. $$ \left[\begin{array}{rrrr} -1 & 4 & 2 & -7 \\ 0 & 5 & -3 & 6 \\ 0 & 0 & -5 & 1 \\ 0 & 0 & 0 & 11 \end{array}\right] $$
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