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Chapter 7: Q12E (page 336)

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.

[2-2001-100003-4002-3]

Short Answer

Expert verified

Eigenvalues are:

位1=0,almuu(0)=1位2=-1,almu(-1)=2位4=1,almu(1)=1

Step by step solution

01

Eigenvalues

  • In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by 位, is the factor by which the eigenvector is scaled.
  • Eigenvalues of a triangular matrix are its diagonal matrix.
02

Step 2: Finding all real eigenvalues, with their algebraic multiplicities

Since, given matrix is triangular its eigenvalues arethe entries on the main diagonal.

Find eigenvalues using characteristic equation as:

det(A-位l)=02-位-2001-1-位00003-位-4002-3=02-位-1-位+23-位-3-位+8=0

位=0,位2,3=-1,位4=1

Hence, the answer is:

位1=0,almuu(0)=1位2=-1,almu(-1)=2位4=1,almu(1)=1

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Most popular questions from this chapter

If 0 is an eigenvalue of a matrix A, then det A = 0.

Consider a rotationT(x鈫�)=Ax鈫�in鈩�3in. (That is, A is an orthogonal 3x3matrix with determinant 1.) Show that T has a non-zero fixed point [i.e., a vectorT(x鈫�)=Ax鈫�in鈩�3withT(v鈫�)=v鈫�]. This result is known as Euler鈥檚 theorem, after the great Swiss mathematician Leonhard Euler (1707鈥�1783). Hint: Consider the characteristic polynomialrole="math" localid="1659595800447" fA(位). Pay attention to the intercepts with both axes. Use Theorem 7.1.4.

There exists a real 5 脳 5 matrix without any real eigenvalues.

Is v鈬赌an eigenvector of A3? If so, what is the eigenvalue?

(a). If 2i is an eigenvalue of a real 2 脳 2 matrix A, findA2.

(b). Give an example of a real 2 脳 2 matrix A such that all the entries of A are nonzero and 2i is an eigenvalue of A. ComputeA2and check that your answer agrees with part (a).
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