Need help on understanding this notation

[Firepanda]

Where it says max_Q I don't understand.

Q is a matrix, so are A and P.

I know what q₁ and the 2-norm of P are, but what on Earth does max Q mean?

I started with the matrix A, I've selected Q as positive definite arbitrarily, and I received a matrix P. I need to now find rho but I'm stuck on what max Q means.

 

[n!kofeyn]

max_Q means to take the maximum over all possible Q that satisfy the given equation. q₁ and P will vary depending on what the matrix Q is.

 

[Firepanda]

max_Q means to take the maximum over all possible Q that satisfy the given equation. q₁ and P will vary depending on what the matrix Q is.

I still really don't understand this part, can you simplify it a bit?

Say if I have a matrix Q = [1 2; 3 4]

What is the max Q?

 

[HallsofIvy]

I still really don't understand this part, can you simplify it a bit?

Say if I have a matrix Q = [1 2; 3 4]

What is the max Q?

There is no such thing. Matrices are not "ordered" and so do not have "max" or "min".

It's not "max Q", it is $$\begin{array} amax \\ Q\end{array}$$, the maximum taken by a numerical expression over all possible values of Q. This isn't saying anything about a maximum of Q, it's talking about the maximum value of the numerical expression $q_1/||P||$ where Q can be any matrix and P must be such that $A^TP+ PA+ Q= 0$

 

[Firepanda]

There is no such thing. Matrices are not "ordered" and so do not have "max" or "min".

It's not "max Q", it is $$\begin{array} amax \\ Q\end{array}$$, the maximum taken by a numerical expression over all possible values of Q. This isn't saying anything about a maximum of Q, it's talking about the maximum value of the numerical expression $q_1/||P||$ where Q can be any matrix and P must be such that $A^TP+ PA+ Q= 0$

Ok thanks, so I don't need to alter the calculation of $q_1/(2||P||)$ and treat it the same as if $$\begin{array} amax \\ Q\end{array}$$ wasn't there?