/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 If \(A\) is an invertible matrix... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 14

If \(A\) is an invertible matrix that is orthogonally diagonalizable, show that \(A^{-1}\) is orthogonally diagonalizable.

Short Answer

Expert verified
Yes, because \( A^{-1} = PD^{-1}P^T \) maintains orthogonal diagonalization.

Step by step solution

01

Understand Orthogonal Diagonalization

If a matrix \( A \) is orthogonally diagonalizable, it means there exists an orthogonal matrix \( P \) and a diagonal matrix \( D \) such that \( A = PDP^T \). The matrix \( P \) contains the orthonormal eigenvectors of \( A \), and \( D \) has the corresponding eigenvalues on its diagonal.
02

Express the Inverse Matrix

Since \( A \) is invertible and orthogonally diagonalizable, it can be expressed as \( A = PDP^T \). To find \( A^{-1} \), use the property of orthogonal matrices where \( P^T = P^{-1} \). Thus, we have \( A^{-1} = (PDP^T)^{-1} = (P^T)^{-1} D^{-1} P^{-1} = PD^{-1}P^T \).
03

Check the Properties of the Inverse

The diagonal matrix \( D^{-1} \) is obtained by taking the reciprocal of the non-zero eigenvalues on the diagonal of \( D \), because \( A \) is invertible. As \( D \) is diagonal, so is \( D^{-1} \). Furthermore, because \( P \) is orthogonal, the expression \( PD^{-1}P^T \) preserves the orthogonal diagonalization property.
04

Conclusion

Since \( A^{-1} = PD^{-1}P^T \) with \( P \) being orthogonal and \( D^{-1} \) being diagonal, \( A^{-1} \) is orthogonally diagonalizable. This concludes that the inverse of an orthogonally diagonalizable matrix is also orthogonally diagonalizable.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Invertible Matrix
An invertible matrix is a square matrix that possesses an inverse. This means if you have a matrix \( A \), there exists another matrix, denoted as \( A^{-1} \), such that when you multiply them together, you retrieve the identity matrix \( I \). The identity matrix is essentially a diagonal matrix with ones on its diagonal and zeros elsewhere. Here are some key points about invertible matrices:
  • An invertible matrix must have a non-zero determinant.
  • Only square matrices can be invertible; matrices with different numbers of rows and columns cannot.
  • If you perform row operations on a matrix and can reduce it to the identity matrix, then the original matrix is invertible.
The importance of invertible matrices is seen across various mathematical applications as they allow for the solution of linear equations and transformations.
Eigenvalues
Eigenvalues are special numbers associated with a matrix that provide vital insights into its properties. If you transform a matrix \( A \) by its eigenvector, the result is a scalar multiplied by the original vector. This scalar is termed the eigenvalue. The fundamental equation for finding eigenvalues is:\[ Av = \lambda v \]where \( A \) is the matrix, \( v \) is the eigenvector, and \( \lambda \) is the eigenvalue. Understanding eigenvalues involves:
  • Calculating them by solving the characteristic polynomial, which is derived from \( \text{det}(A - \lambda I) = 0 \).
  • Recognizing that they reveal important traits about the matrix, such as stability and vibration modes in engineering contexts.
  • Noting that for an orthogonally diagonalizable matrix, all eigenvalues are real.
They are essential in areas like system stability analysis and quantum mechanics, acting as a bridge between a matrix and its functional transformations.
Orthogonal Matrix
An orthogonal matrix is a square matrix \( P \) whose inverse equals its transpose, meaning \( P^T = P^{-1} \). This property greatly simplifies various computations. Key features include:
  • All columns (and rows) are orthonormal vectors, meaning they are perpendicular and of unit length.
  • The product of an orthogonal matrix with its transpose yields the identity matrix: \( PP^T = I \).
  • Orthogonal matrices maintain the lengths and angles in transformations, which makes them particularly valuable in preserving geometric structures.
The beauty of orthogonal matrices lies in their ability to diagonalize symmetric matrices through orthogonal diagonalization, preserving critical information such as eigenvalues and eigenvectors.
Diagonal Matrix
Diagonal matrices are remarkably simple yet powerful in their structure. A diagonal matrix \( D \) is characterized by having non-zero entries solely on its diagonal and zeros elsewhere, making matrix operations particularly efficient. Consider these benefits:
  • Multiplying a diagonal matrix by a vector straightforwardly scales each component of the vector by the corresponding diagonal element.
  • Inverse of a diagonal matrix, if existent, is also diagonal鈥攃ompute it by taking the reciprocal of each non-zero diagonal element.
  • They simplify calculations of powers of matrices since harming a diagonal matrix is simply raising each of its diagonal components to that power.
This ease of computation makes diagonal matrices essential in simplifying complex linear transformations and solving differential equations, allowing for clearer insights and streamlined calculations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Orthogonally diagonalize the matrices by finding an orthogonal matrix Q and a diagonal matrix \(D\) such that \(Q^{T} A Q=D.\) \(A=\left[\begin{array}{llll}1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\\ 0 & 0 & 1 & 1\end{array}\right]\)

Find the symmetric matrix \(A\) associated with the given quadratic form. \(3 x^{2}-3 x y-y^{2}\)

Sometimes the graph of a quadratic equation is a straight line, a pair of straight lines, or a single point. We refer to such a graph as a degenerate conic. It is also possible that the equation is not satisfied for any values of the variables, in which case there is no graph at all and we refer to the conic as an imaginary conic. Identify the conic with the given equation as either degenerate or imaginary and, where possible, sketch the graph. $$x^{2}-2 x y+y^{2}+2 \sqrt{2} x-2 \sqrt{2} y=0$$

Find the symmetric matrix \(A\) associated with the given quadratic form. \(x_{1}^{2}-x_{3}^{2}+8 x_{1} x_{2}-6 x_{2} x_{3}\)

Classify each of the quadratic forms as positive definite, positive semidefinite, negative definite, negative semidefinite, or indefinite \(x_{1}^{2}+2 x_{2}^{2}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.