Could you prove that f(A)>0 whenever A>0?

Question

A is a 3x3 matrix. A is called positive, denoted A>0, if every entry of A is positive. Let $$f(x)=x^6 - x^3 + 1$$ Could you prove that f(A)>0 whenever A>0?

Answer 1

Idea said: Spoiler counterexample \(\displaystyle A=\left( \begin{array}{ccc} u & u & u \\ u & u & u \\ u & u & u \end{array} \right)\) where \(\displaystyle 00 whenever A>0. Hopefully you could help me figure one such polynomial. Thank you in advance

Answer 2

Idea said: Spoiler counterexample \(\displaystyle A=\left( \begin{array}{ccc} u & u & u \\ u & u & u \\ u & u & u \end{array} \right)\) where \(\displaystyle 0

Answer 3

Spoiler counterexample \(\displaystyle A=\left( \begin{array}{ccc} u & u & u \\ u & u & u \\ u & u & u \end{array} \right)\) where \(\displaystyle 0

Answer 4

Plato said: Actually \(\displaystyle f(A)=A^6 - A^3 + 1\) is an abuse of notation. If it were \(\displaystyle A^6 - A^3\) that is possible. SEE THIS How do you understand how to use \(\displaystyle +1~?\) Click to expand... It was my mistake/typo. It should be $A^6-A^3+I$ where $I$ is the 3x3 identity matrix.

Answer 5

MathTutoringByDrLiang said: Yes, A stands for an arbitrary 3x3 matrix. We say A>0 (A is then called positive) if every entry of A is positive, but I didn't say A has a numerical value. Furthermore, f(A) means A^6 - A^3 + 1. Click to expand... Actually \(\displaystyle f(A)=A^6 - A^3 + 1\) is an abuse of notation. If it were \(\displaystyle A^6 - A^3\) that is possible. SEE THIS How do you understand how to use \(\displaystyle +1~?\)

Answer 6

Yes, A stands for an arbitrary 3x3 matrix. We say A>0 (A is then called positive) if every entry of A is positive, but I didn't say A has a numerical value. Furthermore, f(A) means A^6 - A^3 + 1. Hopefully I have answered your question.

Answer 7

MathTutoringByDrLiang said: A is a 3x3 matrix. A is called positive, denoted A>0, if every entry of A is positive. Let $$f(x)=x^6 - x^3 + 1$$ Could you prove that f(A)>0 whenever A>0? Click to expand... "A" is supposed to be the name of a 3 x 3 matrix. How can it also have a numerical value at the same time!?