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Chapter 7: Problem 42

Proof Prove that if matrix \(A\) is diagonalizable, then \(A^{T}\) is diagonalizable.

Short Answer

Expert verified
If a matrix \(A\) is diagonalizable, then \(A^T\) is also diagonalizable. The reason for this is that transposing a matrix does not alter its eigenvalues, and thus the condition for diagonalizability (the equality of algebraic and geometric multiplicity for each eigenvalue) remains unchanged after transposition.

Step by step solution

01

Understand the concept of diagonalizable matrix

A square matrix \(A\) is said to be diagonalizable if it is similar to a diagonal matrix. This means that there exists an invertible matrix \(P\) such that if \(D\) is a diagonal matrix, it holds that \(A = PDP^{-1}\). Also, it's important to remember that the diagonal elements of \(D\) are the eigenvalues of \(A\).
02

Recall that transposes preserve eigenvalues

The eigenvalues of a matrix and its transpose are the same. Let \(A\) be a square matrix with eigenvalues \(\lambda_1, \(\lambda_2\), ..., \(\lambda_n\). Then the eigenvalues of \(A^T\) are also \(\lambda_1, \(\lambda_2\), ..., \(\lambda_n\).
03

Show that the algebraic and geometric multiplicities remain the same

The geometric multiplicity of an eigenvalue \(\lambda\) is the dimension of the corresponding eigenspace. If the geometric multiplicity and the algebraic multiplicity of each eigenvalue of a matrix \(A\) are equal then \(A\) is diagonalizable. Therefore, since the transpose operation does not change the eigenvalues, it also does not change the algebraic and geometric multiplicities. Hence, if \(A\) is diagonalizable then so is \(A^T\).

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Key Concepts

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

Eigenvalues
Eigenvalues are fundamental in the study of matrices and are pivotal in matrix diagonalization. To put it simply, an eigenvalue is a special number associated with a matrix. If we have a square matrix \(A\), an eigenvalue is denoted by \(\lambda\) and satisfies the equation \(A\mathbf{v} = \lambda\mathbf{v}\). Here, \(\mathbf{v}\) is a non-zero vector known as an eigenvector.
  • Eigenvalues can be found by solving the characteristic equation of a matrix, which is obtained from \(\det(A - \lambda I) = 0\).
  • Each eigenvalue can have one or more corresponding eigenvectors, forming an eigenspace.
  • In diagonalization, these eigenvalues become the diagonal elements of the diagonal matrix \(D\).
Understanding eigenvalues helps us to determine important matrix properties and simplifies complex matrix operations.
Geometric Multiplicity
Geometric multiplicity provides insights into the structure of the eigenvectors associated with an eigenvalue. For a given eigenvalue \(\lambda\), its geometric multiplicity is the number of linearly independent eigenvectors corresponding to \(\lambda\).
  • Mathematically, it is the dimension of the eigenspace formed by the eigenvectors of \(\lambda\).
  • The geometric multiplicity of an eigenvalue is always less than or equal to its algebraic multiplicity.
  • If the geometric multiplicity of each eigenvalue equals its algebraic multiplicity, the matrix is diagonalizable.
Hence, the geometric multiplicity plays a crucial role in determining if a matrix can be simplified into a diagonal form.
Algebraic Multiplicity
The algebraic multiplicity of an eigenvalue refers to how many times that eigenvalue appears as a root of the characteristic polynomial of a matrix. For a given eigenvalue \(\lambda\), its algebraic multiplicity is represented as \(m\), where \(\det(A - \lambda I) = 0\).
  • Algebraic multiplicity is always a positive integer.
  • An eigenvalue's algebraic multiplicity indicates its repeat count in the diagonal of a diagonal matrix \(D\) when matrix \(A\) is diagonalized.
  • A higher algebraic multiplicity does not guarantee more unique eigenvectors.
Algebraic multiplicity, when compared with geometric multiplicity, helps in determining if matrix \(A\) can be diagonalized.
Transpose of a Matrix
The transpose of a matrix is an operation that flips a matrix over its diagonal. This means that the row entries become column entries and vice versa. If you have a matrix \(A\) with elements \(a_{ij}\), then the transpose matrix \(A^T\) has elements \(a_{ji}\).
  • Transposing a matrix does not change its eigenvalues. If \(\lambda\) is an eigenvalue of \(A\), it is also an eigenvalue of \(A^T\).
  • The process of obtaining a transpose is straightforward: just interchange the rows and columns.
  • Even though eigenvalues are preserved, the corresponding eigenvectors may be different.
Understanding matrix transposition is crucial as it relates closely to many matrix operations and can simplify computations in various areas of linear algebra.

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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.

Showing That a Matrix Is Not Diagonalizable In Exercises \(15-22,\) show that the matrix is not diagonalizable. $$ \left[\begin{array}{rrr} 1 & -2 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{array}\right] $$

Writing Can a matrix be similar to two different diagonal matrices? Explain.

Guided Proof Prove that nonzero nilpotent matrices are not diagonalizable. Getting Started: From Exercise 80 in Section 7.1 you know that 0 is the only eigenvalue of the nilpotent matrix \(A\). Show that it is impossible for \(A\) to be diagonalizable. (i) Assume \(A\) is diagonalizable, so there exists an invertible matrix \(P\) such that \(P^{-1} A P=D,\) where \(D\) is the zero matrix. (ii) Find \(A\) in terms of \(P, P^{-1},\) and \(D\) (iii) Find a contradiction and conclude that nonzero nilpotent matrices are not diagonalizable.

For an invertible matrix \(A,\) prove that \(A\) and \(A^{-1}\) have the same eigenvectors. How are the eigenvalues of \(A\) related to the eigenvalues of \(A^{-1} ?\)
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