# Why Are Some System States Indistinguishable in Observability Analysis?

• Engineering
• pj33
This is why the vectors in the null space are considered states, but differ from the definition of a state vector.
pj33
Homework Statement
Why the null space of the Observable matrix is not the same as the state vector?
Relevant Equations
None
https://en.wikipedia.org/wiki/Observability

I am studying observability and I try to get some intuition on the topic.
Given the observable matrix, we can find the null space. However, the vectors of the null space are states but this differs from the definition of what a state vector of a system is. In addition, the null space indicates that all the states with the same velocity are indistinguishable. Why is this the case ?

Why is that?
Is it because the state vector is a vector containing other vectors (position velocity etc), that's why it differs from the state in the null space.

The state vector of a system usually contains information about the position, velocity, and other physical properties of the system. The null space of a matrix contains vectors that can not be expressed as linear combinations of the columns of the matrix. In the case of an observable matrix, this means that there are certain states that cannot be described in terms of the variables used to construct the matrix. These states are indistinguishable from one another, because they cannot be distinguished by the observations available from the matrix.

## 1. What are unobservable states?

Unobservable states refer to variables or conditions that cannot be directly measured or observed in a scientific experiment or study. These states are often inferred from observable data or indirectly measured through other means.

## 2. Why are unobservable states important in science?

Unobservable states play a crucial role in scientific research as they help explain and predict observable phenomena. They also provide a deeper understanding of complex systems and allow for the development of theories and models.

## 3. How do scientists study unobservable states?

Scientists use various methods to study unobservable states, such as statistical analysis, mathematical modeling, and indirect measurements. They also rely on theories and hypotheses to make predictions and test their validity through experiments or observations.

## 4. Can unobservable states be proven?

No, unobservable states cannot be proven in the same way that observable facts can be. However, scientists can provide evidence and support for the existence of unobservable states through rigorous research and experimentation.

## 5. What are some examples of unobservable states?

Examples of unobservable states include emotions, thoughts, beliefs, and attitudes. Other examples can include gravity, dark matter, and quantum particles, which cannot be directly observed but are inferred through their effects on observable phenomena.

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