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Chapter 7: Problem 42

In Exercises 39-42, determine whether the matrix is orthogonally diagonalizable. $$\left[\begin{array}{rrrr} 0 & 1 & 0 & -1 \\ 1 & 0 & -1 & 0 \\ 0 & -1 & 0 & -1 \\ -1 & 0 & -1 & 0 \end{array}\right]$$

Short Answer

Expert verified
The given matrix is not orthogonally diagonalizable.

Step by step solution

01

Checking matrix symmetry

The given matrix is: \[ \left[\begin{array}{rrrr} 0 & 1 & 0 & -1 \\ 1 & 0 & -1 & 0 \\ 0 & -1 & 0 & -1 \\ -1 & 0 & -1 & 0 \end{array}\right] \]. We will check if the matrix is symmetric by comparing the elements. A square matrix is symmetric if all elements satisfy the condition that the ith row, jth column element is the same as the jth row, ith column element.
02

Comparing matrix elements

Comparing the elements of the matrix, we find that the matrix is not symmetric. For example, the element at the 2nd row, 3rd column is -1, but the element at the 3rd row, 2nd column is -1. Thus, the given matrix is not symmetric.
03

Final Conclusion

Since the given matrix is not symmetric, we can conclude that the matrix is not orthogonally diagonalizable. A matrix must be symmetric in order to be orthogonally diagonalizable.

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Key Concepts

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

Orthogonally Diagonalizable
To determine if a matrix is orthogonally diagonalizable, we check for two main conditions. Firstly, the matrix must be square, meaning it has the same number of rows and columns. Secondly, the matrix must be symmetric. Orthogonal diagonalization involves expressing a matrix as a product of an orthogonal matrix and a diagonal matrix. Symmetric matrices always have a full set of orthogonal eigenvectors and therefore can be orthogonally diagonalized. In solving the exercise problem, we find that the matrix given doesn't meet these conditions, meaning it is not eligible for orthogonal diagonalization.
Symmetric Matrix
A symmetric matrix holds a distinct property: it is mirrored or symmetrical across its main diagonal. This implies that for a square matrix, the element at the ith row and jth column is equal to the element at the jth row and ith column. Mathematically, this symmetrically translates to \( a_{ij} = a_{ji} \). When working on the given exercise matrix, this condition is critical because, without symmetry, a matrix cannot be orthogonally diagonalized. Despite the matrix in the exercise being square, it fails the symmetry test, highlighting one of the crucial tests to apply when considering diagonalization.
Matrix Symmetry
Matrix symmetry is a foundational concept in linear algebra that not only relates to the aesthetic organization of the matrix but also to its mathematical properties. Symmetric matrices, for instance, guarantee real eigenvalues. In the case of the exercise matrix, we test for symmetry by flipping the matrix elements around the main diagonal to see if the original matrix still holds. The lack of this symmetry in our matrix confirms its ineligibility for orthogonal diagonalization and stresses the importance of matrix symmetry in determining mathematical characteristics and operations.
Linear Algebra Concepts
Linear algebra involves many critical concepts, of which understanding matrix properties is fundamental. Some key aspects include:
  • Matrix operations: Including addition, multiplication, transposition, and inversion.
  • Eigenvalues and eigenvectors: Important for understanding linear transformations and stability.
  • Orthogonal and diagonalizable matrices: Central in simplifying matrix operations and studying matrix behavior.
These concepts form a toolbox to address and analyze matrices efficiently. In exercises such as the one provided, identifying whether a matrix is symmetric is a step deeply rooted in these principles, aiding in comprehending broader linear algebra operations and theories.

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Most popular questions from this chapter

(a) Explain how to model population growth using an age transition matrix and an age distribution vector, and how to find a stable age distribution vector. (b) Explain how to use a matrix equation to solve a system of first-order linear differential equations. (c) Explain how to use the Principal Axes Theorem to perform a rotation of axes for a conic and a quadric surface. (d) Explain how to solve a constrained optimization problem.

Find the matrix \(A\) of the quadratic form associated with the equation. Then find the eigenvalues of \(A\) and an orthogonal matrix \(P\) such that \(P^{T} A P\) is diagonal. $$13 x^{2}+6 \sqrt{3} x y+7 y^{2}-16=0$$

Calculus For a real number \(x,\) you can define \(e^{x}\) by the series \(e^{x}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\cdots\) In a similar way, for a square matrix \(X,\) you can define \(e^{x}\) by the series \(e^{x}=I+X+\frac{1}{2 !} X^{2}+\frac{1}{3 !} X^{3}+\frac{1}{4 !} X^{4}+\cdots\) Evaluate \(e^{x},\) where \(X\) is the square matrix shown. (a) \(X=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) (b) \(X=\left[\begin{array}{ll}1 & 0 \\ 1 & 0\end{array}\right]\) (c) \(X=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\) (d) \(x=\left[\begin{array}{rr}2 & 0 \\ 0 & -2\end{array}\right]\)

Find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. $$z=5 x^{2}+12 x y+5 y^{2} ; x^{2}+y^{2}=1$$

Use your school's library, the Internet, or some other reference source to find real-life applications of constrained optimization.
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