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Chapter 7: Q14E (page 336)

Consider a 4 脳 4 matrixA=|BC0D|where B, C, and D are 2 脳 2 matrices. What is the relationship among the eigenvalues of A, B, C, and D?

Short Answer

Expert verified

Every eigenvalue of A is eigenvalue of B and also of D.

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 \lambda, is the factor by which the eigenvector is scaled.

02

Step 2: Finding values of k for which A have two distinct eigenvalues

We can clearly see that,

detA-位l4=0detB-位l2C0D-位l2=0detB-位l2detD-位l2=0

Hence, Every eigenvalue of A is an eigenvalue of B and also of D.

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