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Chapter 7: Q18E (page 336)

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

Short Answer

Expert verified

Eigenvalue of A=位1,2=a卤b

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 , is the factor by which the eigenvector is scaled.

02

Step 2: Find all eigenvalues of A

We can clearly see that,

detA-位l=0a-位bb-a-位=0a-位-b2=0位1,2=2a卤4a2-4a2-b22=a卤b

Hence, 位1,2=a卤b.

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Most popular questions from this chapter

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.

Consider an upper triangular n脳nmatrix Awithaij鈮�0fori=1,2,鈥�,mandaij=0fori=m+1,鈥�n. Find the algebraic multiplicity of the eigenvalueof. Without using Theorem 7.3.6, what can you say about the geometric multiplicity?

TRUE OR FALSE

18. If A and B are nxn matrices, if 伪is an eigenvalue of A, and if 尾is an eigenvalue of B, then 伪尾 must be an eigenvalue of AB.

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?

Find a 2脳2matrix A such that [31]and [12] are eigenvectors of A , with eigenvalues 5 and 10 , respectively.
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