4. (a) Show that if A₁,..., Ak are square matrices (not necessarily of the same size), and A = = diag(A₁,. , Ak), then k rank(A) = rank(A₁), nullity(A) = nullity (A₁). (b) Let A be a Jordan matrix. Use (a) to show that for any λ = F, nullity (A - XI) = the number of Jordan blocks in A with diagonal entries X. k i=1 i=1 (c) Use (b) to show that a Jordan matrix is not diagonalizable unless it is already diagonal. (Hint: Recall from Chapter 5 that a matrix is diagonalizable if and only if ...)
4. (a) Show that if A₁,..., Ak are square matrices (not necessarily of the same size), and A = = diag(A₁,. , Ak), then k rank(A) = rank(A₁), nullity(A) = nullity (A₁). (b) Let A be a Jordan matrix. Use (a) to show that for any λ = F, nullity (A - XI) = the number of Jordan blocks in A with diagonal entries X. k i=1 i=1 (c) Use (b) to show that a Jordan matrix is not diagonalizable unless it is already diagonal. (Hint: Recall from Chapter 5 that a matrix is diagonalizable if and only if ...)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![... 9
Ak are square matrices (not necessarily of the same size),
, Ak), then
4. (a) Show that if A₁,
and A = diag(A₁,
k
rank(A) = Σ rank(A₁), nullity(A) =
i=1
k
nullity (A₂).
(b) Let A be a Jordan matrix. Use (a) to show that for any λ E F,
nullity(A - XI) = the number of Jordan blocks in A with diagonal entries X.
i=1
(c) Use (b) to show that a Jordan matrix is not diagonalizable unless it is already
diagonal. (Hint: Recall from Chapter 5 that a matrix is diagonalizable if and
only if ...)
(d) Let T: V → V be a linear operator on a finite-dimensional vector space V
and let 3 be a Jordan basis for T. Use (c) to show that T is diagonalizable if
and only if [T] is diagonal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2e948f6-fd6f-485f-942e-c931230f8579%2F2df504b2-16a2-4208-b5a7-5f61889fbc4c%2Fl2zm9i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:... 9
Ak are square matrices (not necessarily of the same size),
, Ak), then
4. (a) Show that if A₁,
and A = diag(A₁,
k
rank(A) = Σ rank(A₁), nullity(A) =
i=1
k
nullity (A₂).
(b) Let A be a Jordan matrix. Use (a) to show that for any λ E F,
nullity(A - XI) = the number of Jordan blocks in A with diagonal entries X.
i=1
(c) Use (b) to show that a Jordan matrix is not diagonalizable unless it is already
diagonal. (Hint: Recall from Chapter 5 that a matrix is diagonalizable if and
only if ...)
(d) Let T: V → V be a linear operator on a finite-dimensional vector space V
and let 3 be a Jordan basis for T. Use (c) to show that T is diagonalizable if
and only if [T] is diagonal.
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