invertible matrix PE GLn(C) such that B := P-¹AP is strictly upper triangular, as follows. For 1 ≤ i ≤n, let x; denote the standard basis column vector of length n, with 1 in the i-th position and 0 elsewhere. (a) Show that for any invertible matrix P we have that A is nilpotent if and only if P-¹AP is nilpotent. (b) Show that for any strictly upper triangular matrix B, we have that Bx₁ 0 and for any i 1 the vector Bx, is a linear combination of ₁,...,xi-1. Conclude that B x = 0 for every 1 ≤ i ≤n and therefore B = 0. = (c) Assuming that A is nilpotent with index of nilpotency m≥ 1, show that there exists a non-zero vector y such that Ay = 0. (Hint, consider the column vectors of Am-10 and the fact that A. Am-1 = = 0.) calx se CL
invertible matrix PE GLn(C) such that B := P-¹AP is strictly upper triangular, as follows. For 1 ≤ i ≤n, let x; denote the standard basis column vector of length n, with 1 in the i-th position and 0 elsewhere. (a) Show that for any invertible matrix P we have that A is nilpotent if and only if P-¹AP is nilpotent. (b) Show that for any strictly upper triangular matrix B, we have that Bx₁ 0 and for any i 1 the vector Bx, is a linear combination of ₁,...,xi-1. Conclude that B x = 0 for every 1 ≤ i ≤n and therefore B = 0. = (c) Assuming that A is nilpotent with index of nilpotency m≥ 1, show that there exists a non-zero vector y such that Ay = 0. (Hint, consider the column vectors of Am-10 and the fact that A. Am-1 = = 0.) calx se CL
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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Transcribed Image Text:7. Show that an n x n matrix A E Matnxn (C) is nilpotent if and only if there exists an
invertible matrix PE GLn (C) such that B = P-¹AP is strictly upper triangular, as
follows. For 1 ≤ i ≤n, let x; denote the standard basis column vector of length n,
with 1 in the i-th position and 0 elsewhere.
(a) Show that for any invertible matrix P we have that A is nilpotent if and only if
P-¹AP is nilpotent.
(b) Show that for any strictly upper triangular matrix B, we have that Bx₁
= 0 and
for any
i > 1 the vector Bx, is a linear combination of x₁,...,xi-1. Conclude
that Brxi
= 0 for every 1 ≤ i ≤n and therefore B = 0.
(c) Assuming that A is nilpotent with index of nilpotency m≥ 1, show that there
exists a non-zero vector y such that Ay = 0. (Hint, consider the column vectors
of Am-10 and the fact that A· Am−¹ = 0.)
(d) Let
:= ker(A²) = {y € Cn | A²y = 0}, and let d₂ = dim(V₂). Show that
{0} = V₁ ≤ V₁ C... CVm = C,
and there exists a basis {y₁,..., yn} for Cn such that {y₁,..., yd,} is a basis for
Ve for each 1 ≤ ≤n. (Hint: you can proceed by induction: given a basis
{₁,...,Yde} for V₁, extend it to a basis for Ve+1.)
(e) Show that for each 2 ≤ ≤ n and each 1 ≤ i ≤ de, Ay; is a linear combination
of Y₁,...,Yde-1. Conclude that for the (change-of-basis) matrix P such that yi
is its i-th column vector we have that P-¹AP = B is strictly upper triangular.
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