Note: This question may seem extremely elementary, but I am not a beginner to signal processing, linear system theory, control theory, etc. I believe my confusion is over a subtle point, so, before responding, please read the entire question so you understand the source of my confusion first.

Consider this simple autonomous system (autonomous means it does not depend on its input, $u$):

$$\dot{x}(t) = A x(t)$$

Assume the output is the state ($y(t) = x(t)$) and $A \neq 0$. My questions are:

1. Is this system necessarily linear?
2. Is this system necessarily time-invariant?
3. What is the zero-input response of this system?
4. Would the answers above change if we assumed the system was causal?

Easy, right?

1. Obviously yes, since it is in the form $\dot{x}(t) = Ax(t)$
2. Obviously yes, since $A$ does not depend on $t$
3. The zero-input response is: $x(t) = x(0)\,e^{At}$
4. No, these are unrelated properties.

But wait!

1. Linearity means a linear combination of the inputs must give the same linear combination of the outputs. If I scale the input, $u$, then $x$ is not affected at all. So how can it be linear?!
2. Time-invariance means that if I delay the input, the output is delayed by the same amount of time. But if I delay the input, $u$, then $x$ is not affected at all. So how can it be time-invariant?!
3. If the initial state is an "input": Doesn't linearity imply $x(0) = 0 \implies x(t) = 0$?
   If the initial state isn't an "input": Isn't the system nonlinear unless $x(0) = 0$? (see #1)
4. Wouldn't zero-input give zero-output for a causal LTI system (the "ZIZO" property)?

Hopefully it's clear why I'm confused.
Where am I going wrong?

Note: This question may seem extremely elementary, but I am not a beginner to signal processing, linear system theory, control theory, etc. I believe my confusion is over a subtle point, so, before responding, please read the entire question so you understand the source of my confusion first.

Consider this simple autonomous system (autonomous means it has no direct dependence on $t$):

$$\dot{x}(t) = A x(t)$$

Assume the output is the state ($y(t) = x(t)$) and $A \neq 0$. My questions are:

1. Is this system necessarily linear?
2. Is this system necessarily time-invariant?
3. What is the zero-input response of this system?
4. Would the answers above change if we assumed the system was causal?

Easy, right?

1. Obviously yes, since it is in the form $\dot{x}(t) = Ax(t)$
2. Obviously yes, since $A$ does not depend on $t$
3. The zero-input response is: $x(t) = x(0)\,e^{At}$
4. No, these are unrelated properties.

But wait!

1. Linearity means a linear combination of the inputs must give the same linear combination of the outputs. If I scale the input, $u$, then $x$ is not affected at all. So how can it be linear?!
2. Time-invariance means that if I delay the input, the output is delayed by the same amount of time. But if I delay the input, $u$, then $x$ is not affected at all. So how can it be time-invariant?!
3. If the initial state is an "input": Doesn't linearity imply $x(0) = 0 \implies x(t) = 0$?
   If the initial state isn't an "input": Isn't the system nonlinear unless $x(0) = 0$? (see #1)
4. Wouldn't zero-input give zero-output for a causal LTI system (the "ZIZO" property)?

Hopefully it's clear why I'm confused.
Where am I going wrong?

Note: This question may seem extremely elementary, but I am not a beginner to signal processing, linear system theory, control theory, etc. I believe my confusion is over a subtle point, so, before responding, please read the entire question so you understand the source of my confusion first.

Consider this simple autonomous system (autonomous means it does not depend on its input, $u$):

$$\dot{x}(t) = A x(t)$$

My questions are:

1. Is this system necessarily linear?
2. Is this system necessarily time-invariant?
3. What is the zero-input response of this system?
4. Would the answers above change if we assumed the system was causal?

Easy, right?

1. Obviously yes, since it is in the form $\dot{x}(t) = Ax(t)$
2. Obviously yes, since $A$ does not depend on $t$
3. The zero-input response is: $x(t) = x(0)\,e^{At}$
4. No, these are unrelated properties.

But wait!

1. Linearity means a linear combination of the inputs must give the same linear combination of the outputs. If I scale the input, $u$, then $x$ is not affected at all. So how can it be linear?!
2. Time-invariance means that if I delay the input, the output is delayed by the same amount of time. But if I delay the input, $u$, then $x$ is not affected at all. So how can it be time-invariant?!
3. If the initial state is an "input": Doesn't linearity imply $x(0) = 0 \implies x(t) = 0$?
   If the initial state isn't an "input": Isn't the system nonlinear unless $x(0) = 0$? (see #1)
4. Wouldn't zero-input give zero-output for a causal LTI system (the "ZIZO" property)?

Hopefully it's clear why I'm confused.
Where am I going wrong?

Note: This question may seem extremely elementary, but I am not a beginner to signal processing, linear system theory, control theory, etc. I believe my confusion is over a subtle point, so, before responding, please read the entire question so you understand the source of my confusion first.

Consider this simple autonomous system (autonomous means it does not depend on its input, $u$):

$$\dot{x}(t) = A x(t)$$

Assume the output is the state ($y(t) = x(t)$) and $A \neq 0$. My questions are:

1. Is this system necessarily linear?
2. Is this system necessarily time-invariant?
3. What is the zero-input response of this system?
4. Would the answers above change if we assumed the system was causal?

Easy, right?

1. Obviously yes, since it is in the form $\dot{x}(t) = Ax(t)$
2. Obviously yes, since $A$ does not depend on $t$
3. The zero-input response is: $x(t) = x(0)\,e^{At}$
4. No, these are unrelated properties.

But wait!

1. Linearity means a linear combination of the inputs must give the same linear combination of the outputs. If I scale the input, $u$, then $x$ is not affected at all. So how can it be linear?!
2. Time-invariance means that if I delay the input, the output is delayed by the same amount of time. But if I delay the input, $u$, then $x$ is not affected at all. So how can it be time-invariant?!
3. If the initial state is an "input": Doesn't linearity imply $x(0) = 0 \implies x(t) = 0$?
   If the initial state isn't an "input": Isn't the system nonlinear unless $x(0) = 0$? (see #1)
4. Wouldn't zero-input give zero-output for a causal LTI system (the "ZIZO" property)?

Hopefully it's clear why I'm confused.
Where am I going wrong?