a) Describe the kernel and the image of Jn. (b) Use this to show that ??Jn is diagonalisable when n≠0 in F. (c) Show that Jn is never diagonalisable when n=0 in F.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Let Jn be the ?×?n×n matrix whose entries are all equal to 1, and let D(λ1,…,λn) be the n×n diagonal matrix whose non-zero entries are λ1,…,λn∈R. Let x_=(x1,…,xn) be a row vector in ℝ?Rn and let x_t be its transpose, a column vector. Consider the linear transformation Jn:Fn→Fn,  and F is an arbitrary field.

(a) Describe the kernel and the image of Jn.

(b) Use this to show that ??Jn is diagonalisable when n≠0 in F.

(c) Show that Jn is never diagonalisable when n=0 in F.



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