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Chapter 7: Q57E (page 325)

If v鈫�1,v鈫�2,鈥�,v鈫�nis an Eigen basis for both AandB, then matricesAandBmust commute.

Short Answer

Expert verified

The given statement is true because, the given matrices A and B compute.

Step by step solution

01

Definition of Eigenvalue

The dedication of a system's eigenvalues and eigenvectors is vital in physics and engineering, in whichit's farcorresponding to matrix diagonalization and takes place in programs as various as balance analysis, rotating frame physics, and tiny oscillations of vibrating systems, to say a few.

02

Determine the given statement is true or false

Each eigenvalue has an eigenvector that corresponds to it (or, in general, a corresponding proper eigenvector and a corresponding left eigenvector; there may be no analogous differenceamong left and proper for eigenvalues).

Let us know that,

S=v1.....vn

Then, the equation is:

AS=伪1v1...伪nvnBS=尾1v1...尾nvn

which gives us:

ABS=伪1尾1v1鈥�伪n尾nvn=BAS

Since S is invertible, leads to AB=BA

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Most popular questions from this chapter

Consider the matrix A=|abba|where aand bare arbitrary constants. Find all eigenvalues of A.

Find all2脳2matrices for whiche鈬赌1is an eigenvector.

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

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For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

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For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

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