Why is the vector (I−A)^(-1x) positive?

Lance Liu

Lance Liu

Answered question

2022-10-21

Why is the vector ( I A ) 1 x positive?
I was reading a text where they make the conclusion that the product ( I A ) 1 x is a vector with positive entries.
Notes:
1)I is the identity n × n matrix.
2)A is a n × n matrix with zeros on the diagonal and non-negative entries such that A < 1, thus I A is invertible.
3)x is a vector with positive entries.

Answer & Explanation

Amadek6

Amadek6

Beginner2022-10-22Added 21 answers

By definition, we have
( I A ) 1 = I + A + A 2 + + A n + .
Next, we see that A k x has positive entries since A k is a matrix with nonnegative entries and x is a positive vector. Hence ( I A ) 1 x is a positive vector.

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