/*! 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 42 Let \(A\) be an invertible matri... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 42

Let \(A\) be an invertible matrix. Prove that if \(A\) is diagonalizable, so is \(A^{-1}\).

Short Answer

Expert verified
If \( A \) is diagonalizable, then \( A^{-1} \) is also diagonalizable.

Step by step solution

01

Definition of Diagonalizable Matrix

A matrix \( A \) is diagonalizable if there exists an invertible matrix \( P \) and a diagonal matrix \( D \) such that \( A = PDP^{-1} \). We start by using this definition for \( A \).
02

Using the Inverse Property

To find the inverse of \( A \), express it using the given relation: \( A = PDP^{-1} \). We know that \( A^{-1} = (PDP^{-1})^{-1} \).
03

Apply Matrix Inversion Formula

The inverse of a product of matrices \((XY)\) is \((YX)^{-1} = X^{-1}Y^{-1}\). Therefore, apply it to \( A^{-1} = (PDP^{-1})^{-1} = PD^{-1}P^{-1} \).
04

Verify Diagonalizability of \( A^{-1} \)

Since \( D \) is a diagonal matrix, \( D^{-1} \) is also a diagonal matrix (provided all elements on the diagonal are non-zero). Therefore, the equation \( A^{-1} = PD^{-1}P^{-1} \) confirms that \( A^{-1} \) is diagonalizable by the same \( P \).
05

Conclusion

Thus, if \( A \) is diagonalizable as \( A = PDP^{-1} \), it follows that \( A^{-1} = PD^{-1}P^{-1} \) is also diagonalizable, as both involve the transformation by the same matrix \( P \).

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, say matrix \( A \), there exists another matrix, denoted as \( A^{-1} \), such that when you multiply \( A \) by \( A^{-1} \), you get the identity matrix. The identity matrix is a special kind of square matrix with ones on its diagonal and zeros elsewhere.
  • If \( A \times A^{-1} = I \) and \( A^{-1} \times A = I \), where \( I \) is the identity matrix, then \( A \) is invertible.
  • An invertible matrix is also sometimes called a non-singular or non-degenerate matrix.
Not all matrices are invertible. A matrix must be square and have a non-zero determinant to be invertible. These properties are crucial for various matrix-related operations in mathematics.
Matrix Inversion
Matrix inversion deals with finding the inverse of a matrix, provided the matrix is invertible. The process is akin to finding the reciprocal of a number in basic arithmetic.
  • To find the inverse of a matrix \( A \), we need a formula or method that rearranges the elements of \( A \) into \( A^{-1} \).
  • In practical terms, this often involves using elementary row operations to transform \( A \) into the identity matrix, applying the same operations to the identity matrix to find \( A^{-1} \).
Understanding the concept of matrix inversion is critical for solving systems of linear equations, performing transformations, and handling various problems in linear algebra and calculus.
Diagonal Matrix
A diagonal matrix is a type of matrix where all the elements outside the main diagonal are zero. This means if you have a diagonal matrix \( D \), it looks like this: \[ \begin{bmatrix} d_{11} & 0 & 0 \ 0 & d_{22} & 0 \ 0 & 0 & d_{33} \end{bmatrix}\]
  • The values \( d_{11}, d_{22}, d_{33}, \ldots \) are the diagonal elements and are the only potentially non-zero elements in the matrix.
  • A diagonal matrix is particularly simple to work with when it comes to operations like addition, multiplication, and finding powers.
One key property of diagonal matrices is that they are easy to invert, provided none of the diagonal elements are zero. The inverse of a diagonal matrix is simply a diagonal matrix where each diagonal element is replaced by its reciprocal.
Matrix Transformation
Matrix transformation is a fundamental concept in linear algebra, which involves using matrices to transform vectors and spaces. When you have a matrix \( A \) and a vector \( \mathbf{v} \), the product \( A\mathbf{v} \) represents a transformation of \( \mathbf{v} \). This change can be anything from rotating, scaling, or otherwise altering the vector.
  • Transformations provide a useful way to understand geometric concepts and are widely used in computer graphics, physics simulations, and more.
  • Linear transformations preserve vector addition and scalar multiplication, which means the entire space changes in a consistent way.
Special matrices, like diagonal matrices, can provide insights into the effects of these transformations. For instance, diagonal matrices scale the basis vectors of a space, showing how powerful transformations can be effected with relatively simple matrix operations.

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

Write out the first six terms of the sequence defined by the recurrence relation with the given initial conditions. $$a_{1}=128, a_{n}=a_{n-1} / 2 \text { for } n \geq 2$$

Which of the stochastic matrices are regular? $$\left[\begin{array}{lll} \frac{1}{2} & 0 & 1 \\ \frac{1}{2} & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$$

Find the general solution to the given system of differential equations. Then find the specific solution that satisfies the initial conditions. (Consider all functions to be functions of \(t .\) $$\begin{array}{ll} x_{1}^{\prime}=x_{1}+x_{2}, & x_{1}(0)=1 \\ x_{2}^{\prime}=x_{1}-x_{2}, & x_{2}(0)=0 \end{array}$$

Write out the first six terms of the sequence defined by the recurrence relation with the given initial conditions. $$b_{0}=1, b_{1}=1, b_{n}=2 b_{n-1}+b_{n-2} \text { for } n \geq 2$$

Find the general solution to the given system of differential equations. Then find the specific solution that satisfies the initial conditions. (Consider all functions to be functions of \(t .\) $$\begin{array}{l} x^{\prime}=2 x-y, \quad x(0)=1 \\ y^{\prime}=-x+2 y, \quad y(0)=1 \end{array}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.