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Chapter 7: Q7-15E (page 383)

TRUE OR FALSE

15. If matrix A is diagonalizable, then its transpose AT must be diagonalizable as well.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Define matrix

A table of numbers and symbols arranged in the form of rows and columns is known as a matrix.

02

Determine whether the given statement is true or false

Consider the given statement, 鈥淚f the matrix A is diagonalizable, then its transpose AT must be diagonalizable as well.鈥�

Since the matrices A and AT have the same characteristic polynomial, it also has the same eigenvalues and also the same geometric and algebraic multiplicities.

Thus, if A is diagonalizable, then AT is also diagonalizable.

Therefore, the given statement is true.

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Most popular questions from this chapter

Consider the matrix A=|abbc| where a, b, and c are nonzero constants. For which values of a, b, and c does A have two distinct eigenvalues?

Question: If a vectorv鈫�is an eigenvector of both AandB, isv鈫� necessarily an eigenvector ofAB?

For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

5.(45-2-2)

For which3脳3 matrices A does there exist a nonzero matrix M Such that AM=MD,where D=[200030004]Give your answer in terms of eigenvalues of A.

For a given eigenvalue, find a basis of the associated eigensspace .use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.

For each of the matrices A in Exercise1 through 20,find all (real) eigenvalues. Then find a basis of each eigenspaces ,and diagonalize A, if you can. Do not use technology.

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