Question

Let A, B be n × n matrices. The following are two incorrect proofs that ABhas the same non-zero eigenvalues as BA. For each, state two things wrong with the proof: (i) We will prove that AB and BA have the same characteristic equation. We have that

det(AB − λI) = det(ABAA−1 − λAA−1) = det(A(BA − λI)A−1)

= det(A) + det(BA − λI) − det(A)

= det(BA − λI)

Hence det(AB − λI) = det(BA − λI), and so AB and BA have the same eigenvalues

(ii) Suppose not. We will prove by contradiction. Let λ,μ be non-zero eigenvalues of AB,BA respectively such that λ ̸= μ. Then we have that for an eigenvector ⃗v

AB⃗v=λ⃗v, BA⃗v=μ⃗v=⇒(AB−BA)⃗v=(λ−μ)⃗v= ⇒ 0 ⃗v = ( λ − μ ) ⃗v

=⇒ λ − μ = 0

=⇒ λ = μ

Hence we have arrived at a contradiction. So AB and BA have the
same eigenvalues.

Answer #1

Let matrices A,B∈Mn×n(R). Show that if A and B are each similar
to some diagonal matrix, and also have the same eigenvectors (but
not necessarily the same eigenvalues), then AB=BA.

Prove the following statements:
a) If A and B are two positive semideﬁnite matrices in IR ^ n ×
n , then trace (AB) ≥ 0. If, in addition, trace (AB) = 0, then AB =
BA =0
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R(A) = R(B), where R(·) denotes the range of a matrix.
(1) Show that R(A + B) is a subspace of R(A).
(2) Is it always true...

Q.Let A and B be n × n matrices such that A = A^2,
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We say two n × n matrices A and B are similar if there is an
invertible n × n matrix P such that
A = PBP^ -1.
a) Show that if A and B are similar n × n matrices, then they
must have the same determinant.
b) Show that if A and B are similar n × n matrices, then they
must have the same eigenvalues.
c) Give an example to show that A and B can be...

Let A, B, and C be n×n matrices of the form A= [c_1...x...c_n],
B= [c_1...y...c_n], and C= [c_1...x+y...c_n] where x, y, and x+y
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det(C)=det(A)+det(B).

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eigenvalues). Hint: For any (n×n) matrix B, we have
det(BT) = det(B). Remark: Note that, however, it is
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eigenvectors!

Let A, B ? Mn×n be invertible matrices. Prove the following
statement: Matrix A is similar to B if and only if there exist
matrices X, Y ? Mn×n so that A = XY and B = Y X.

4.4.3. Suppose A and B are n × n matrices. Prove that, if AB is
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4.4.4.
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Given that A and B are n × n matrices and T is a linear
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(b) If Au = Av and u and v are 2 distinct vectors, then A is not
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(c) If A or B is not invertible, then AB is not invertible.
(d) If T is invertible and T(u) = T(v), then u =...

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