Question

Let \( A \) be a \( 9 \times 9 \) matrix with characteristic polynomial \[ \operatorname{det}(\lambda I-A)=(\lambda-1)^{2}(\lambda-3)^{3}(\lambda-7)^{4} \text {. } \] Without knowing any further information about \( A \), how many different values are possible for the geometric multiplicity of eigenvalue \( \lambda=7 \) ?

          Let \( A \) be a \( 9 \times 9 \) matrix with characteristic polynomial
\[
\operatorname{det}(\lambda I-A)=(\lambda-1)^{2}(\lambda-3)^{3}(\lambda-7)^{4} \text {. }
\]

Without knowing any further information about \( A \), how many different values are possible for the geometric multiplicity of eigenvalue \( \lambda=7 \) ?

Let A be a 9 × 9 matrix with characteristic polynomial

det(λ I-A)=(λ-1)^2(λ-3)^3(λ-7)^4.

Without knowing any further information about A, how many different values are possible for the geometric multiplicity of eigenvalue λ=7 ?

Added by David M.

Differential Equations and Linear Algebra

Stephen W. Goode, Scott A. Annin 4th Edition

Chapter 7

Instant Answer

Solved by Expert Matthew Elliott

Step 1

The geometric multiplicity of an eigenvalue \(\lambda\) of a matrix \(A\) is defined as the dimension of the null space (kernel) of \(A - \lambda I\), where \(I\) is the identity matrix of the same size as \(A\). This is equivalent to the number of linearly Show more…

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Let \( A \) be a \( 9 \times 9 \) matrix with characteristic polynomial \[ \operatorname{det}(\lambda I-A)=(\lambda-1)^{2}(\lambda-3)^{3}(\lambda-7)^{4} \text {. } \] Without knowing any further information about \( A \), how many different values are possible for the geometric multiplicity of eigenvalue \( \lambda=7 \) ?