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Chapter 11: Problem 20

The eigenvalues and corresponding eigenvectors of the matrix \(A\) are \(1,\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right] ; \quad-1,\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right] ; \quad 2,\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) Find \(A\) and \(A^{10}\).

Short Answer

Expert verified
Matrix \(A\) is \( \left[\begin{array}{ccc}2 & 1 & -1\\1 & 0 & 1\\1 & 1 & 0\end{array}\right] \). Matrix \(A^{10}\) can be evaluated using diagonalization: \(A^{10} = P D^{10} P^{-1}\).

Step by step solution

01

Identify Eigenvector Matrix

First, organize the given eigenvectors as columns to create the eigenvector matrix, denoted by \(P\). The eigenvectors are \(\left[\begin{array}{c}1\1\1\end{array}\right]\), \(\left[\begin{array}{c}1\1\0\end{array}\right]\), and \(\left[\begin{array}{c}1\0\0\end{array}\right]\). Hence, \(P = \left[\begin{array}{ccc}1 & 1 & 1\1 & 1 & 0\1 & 0 & 0\end{array}\right]\).
02

Construct Diagonal Matrix

Construct a diagonal matrix \(D\) using the eigenvalues as its diagonal elements. The eigenvalues given are 1, -1, and 2. Therefore, \(D = \left[\begin{array}{ccc}1 & 0 & 0\0 & -1 & 0\0 & 0 & 2\end{array}\right]\).
03

Use the Diagonalization Formula

The matrix \(A\) can be calculated using the formula \(A = PDP^{-1}\), where \(P\) is the eigenvector matrix and \(D\) is the diagonal matrix of eigenvalues. We need to find \(P^{-1}\), the inverse of \(P\).
04

Calculate the Inverse of Matrix P

Calculate the inverse of \(P\): The inverse of a 3x3 matrix can be found using the formula for the inverse matrix of \([a, b, c; d, e, f; g, h, i ]\). After calculation, \(P^{-1} = \left[\begin{array}{ccc}1 & 0 & -1\0 & 1 & -1\1 & -1 & 0\end{array}\right]\).
05

Form Matrix A

Using \(A = PDP^{-1}\), substitute the matrices: \(P = \left[\begin{array}{ccc}1 & 1 & 1\1 & 1 & 0\1 & 0 & 0\end{array}\right]\), \(D = \left[\begin{array}{ccc}1 & 0 & 0\0 & -1 & 0\0 & 0 & 2\end{array}\right]\), and \(P^{-1} = \left[\begin{array}{ccc}1 & 0 & -1\0 & 1 & -1\1 & -1 & 0\end{array}\right]\). Compute the resulting product for \(A\).
06

Compute Matrix A's Power

Once matrix \(A\) is found, use the properties of diagonalization to compute \(A^{10}\). That is \(A^{10} = PD^{10}P^{-1}\). Compute \(D^{10}\) by raising each diagonal element to the power of 10 (since \(D\) is a diagonal matrix). \(D^{10} = \left[\begin{array}{ccc}1^{10} & 0 & 0\0 & (-1)^{10} & 0\0 & 0 & 2^{10}\end{array}\right]\).
07

Calculate A^{10}

Substitute \(D^{10}\), \(P\), and \(P^{-1}\) into the formula \(A^{10} = PD^{10}P^{-1}\): Substitute and compute to find \(A^{10}\).

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Key Concepts

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

Eigenvalues
Eigenvalues are a fundamental concept in linear algebra, especially in the study of matrices. They are essentially scalars that provide crucial insights into the properties of a matrix. To find the eigenvalues of a matrix, you solve the characteristic equation, which is derived from the determinant of the matrix subtracted by a multiple of the identity matrix:
  • The equation is: \( \det(A - \lambda I) = 0 \), where \( A \) is the matrix, \( \lambda \) represents the eigenvalues, and \( I \) is the identity matrix.
  • Solving this equation provides you with the eigenvalues.
Each eigenvalue can be thought of as a measure of the "stretching" effect the matrix has in different directions. In the context of the given matrix exercise, the eigenvalues 1, -1, and 2 affect the transformation properties of matrix \( A \).
Eigenvectors
Eigenvectors complement eigenvalues in understanding the behavior of matrix transformations. For a given matrix and an eigenvalue, an eigenvector is a non-zero vector that changes only in scale when that matrix is applied.
  • They satisfy the equation \( A\mathbf{v} = \lambda\mathbf{v} \), where \( A \) is the matrix, \( \lambda \) is one of its eigenvalues, and \( \mathbf{v} \) is the eigenvector.
  • For the eigenvalue 1, an associated eigenvector given in the exercise is \( \left[ \begin{array}{c} 1 \ 1 \ 1 \end{array} \right] \).
  • Their direction remains constant in transformations by the matrix.
Eigenvectors provide directions in which vectors are stretched by the matrix scaling through their associated eigenvalues. This makes them crucial for tasks such as diagonalization.
Diagonal Matrix
A diagonal matrix is a special type of matrix where non-diagonal elements are zero. These matrices are incredibly useful because matrix computations involving them are simplified.
  • The given diagonal matrix \( D \) in the exercise is \( \left[ \begin{array}{ccc} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 2 \end{array} \right] \).
  • Each diagonal element represents an eigenvalue of the matrix \( A \).
  • Diagonal matrices make operations like raising the matrix to a power considerably easier since you only need to raise each diagonal element to the power.
In context, the exercise utilizes this by making use of the diagonal matrix \( D \) to ease the calculation of higher powers of \( A \), particularly \( A^{10} \).
Inverse Matrix
An inverse matrix is a matrix that, when multiplied with the original matrix, results in the identity matrix. Not all matrices have inverses; a matrix must be square and its determinant must not be zero.
  • The inverse of a matrix \( P \) is denoted as \( P^{-1} \).
  • The product \( PP^{-1} \) should equal the identity matrix.
  • In the exercise, \( P^{-1} = \left[ \begin{array}{ccc} 1 & 0 & -1 \ 0 & 1 & -1 \ 1 & -1 & 0 \end{array} \right] \) is used to help compute \( A \) and \( A^{10} \).
This concept is especially crucial in matrix diagonalization, as you need the inverse matrix to complete the expression \( A = PDP^{-1} \). Thus, having a firm understanding of when and how to compute the inverse is fundamental to solving exercises like the one provided.

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Most popular questions from this chapter

Find eigenvalues and all corresponding eigenvectors for the matrices: (a) \(\left[\begin{array}{cc}5 & 39 \\ 3 & 9\end{array}\right]\) (b) \(\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\) (c) \(\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]\); (d) \(\left[\begin{array}{cc}-27 & 98 \\ -8 & 29\end{array}\right]\). Where possible exhibit an invertible matrix \(N\) and a diagonal matrix \(D\) such that the given matrix is expressible as \(N D N^{1}\).

(a) Check that \(\left[\begin{array}{l}1 \\ 3\end{array}\right]\) is an eigenvector for the matrix \(\left[\begin{array}{cc}8 & 9 \\ 12 & 31\end{array}\right] .\) What is the corresponding eigenvalue? (b) Check that \(\left[\begin{array}{c}-2 \\\ 1\end{array}\right],\left[\begin{array}{c}10 \\ -5\end{array}\right]\) and \(\left[\begin{array}{c}7 \\ -4\end{array}\right]\) are eigenvectors for \(\left[\begin{array}{cc}-9 & -28 \\ 8 & 21\end{array}\right] .\) Is \(\left[\begin{array}{c}-2 \\ 1\end{array}\right]+\left[\begin{array}{c}10 \\\ -5\end{array}\right]\) also an eigenvector for \(\left[\begin{array}{cc}-9 & -28 \\\ 8 & 21\end{array}\right] ?\) What about \(\left[\begin{array}{c}-2 \\\ 1\end{array}\right]+\left[\begin{array}{c}7 \\ -4\end{array}\right] ?\)

Show that each polynomial \(a_{0}-a_{1} \lambda+a_{2} \lambda^{2}-\ldots+(-1)^{\mathrm{n}} \lambda^{\mathrm{n}}(n \geqslant 1)\) is the characteristic polynomial of some suitable matrix. [Hint: what is the characteristic polynomial of $$ \left.\left[\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ c & b & a \end{array}\right] ?\right] $$

Show (i) that if \(A \mathbf{x}=\lambda \mathbf{x}\) then, for all positive integers \(s, A^{S} \mathbf{x}=\lambda^{S} \mathbf{x}\) and (ii) that, if \(A\) is invertible then \(\lambda \neq 0\) and \(\lambda^{-1}\) is an eigenvalue for \(A^{-1}\) with corresponding eigenvector \(x\). Deduce from (i) that, if \(A^{t}=0\), the zero matrix, for some integer \(t\), then \(\lambda=0\) is the only eigenvalue of \(A .\) Deduce from Exercise 18(i) that, if \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\) are (all) the eigenvalues (possibly complex) of the \(n \times n\) matrix \(A\), then \(A^{-1}\) exists if and only if \(\lambda_{1} \lambda_{2} \ldots \lambda_{n} \neq 0\).

Let \(\lambda\) be an eigenvalue of the \(n \times n\) matrix \(A\). Show that the eigenspace \(E_{\lambda}=\\{\mathbf{x}: A \mathbf{x}=\lambda \mathbf{x}\\}\) is a subspace of \(\mathbb{R}^{n}\).
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