Can you show me please trace|a><b|=<b|a>

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A projection operator is a linear operator P that satisfies P^2=P. In other words, when P acts on a vector, it projects the vector onto a subspace defined by P, and then projects it again onto the same subspace, resulting in the original vector. This means that P^2 is equivalent to P, and thus P has eigenvalues of either 0 or 1. This can be seen by considering the eigenvalue equation P|v>=λ|v>, and applying P again to both sides to get P^2|v>=λP|v>. Since P^2=P, this becomes P|v>=λP|v>, which implies that either λ=1 or λ=0. Therefore,
  • #1
tchem
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Can you show me please trace|a><b|=<b|a> . Thank you?
 
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  • #2
Projector

How can we show that the eigenvalues of a projector are 1 or 0? Thank you
 
  • #3
Unitary

How can we show that the eigenvalues of a unitary operator are complex numbers of unit norm. Thank you
 
  • #4
You should really show an attempt at the question, but:

Tr[ |a><b| ] = sum <ci|a><b|ci>=sum<b|ci><ci|a>=<b|a>
 
  • #5
Anti-Hermition

A is anti-hermition;
please show that <b|A|b> is a pure imaginary number for any |b> in the state space
 
  • #6
Trace

How can we show;
trace A= sum<ei|A|ei>=sum<fi|A|>fi
 
  • #7
tchem, as christianjb points out, you MUST show your own work so far, in order for us to be of help with your homework and coursework. We do not do your work for you here on the PF. Please post your work and thoughts so far for each of the multiple questions you are asking.
 
  • #8
actually, I solved the questions of unitary and anti-hermition, but I cannot prove that the eigenvalues of projector can be 0 or 1 . I get the point and guess it from the exact definition of projection operator however I cannot get it into writing mathematically
 
  • #9
tchem said:
actually, I solved the questions of unitary and anti-hermition, but I cannot prove that the eigenvalues of projector can be 0 or 1 . I get the point and guess it from the exact definition of projection operator however I cannot get it into writing mathematically

What is the exact definition of a projection operator?
 

1. What is a trace in scientific terms?

A trace refers to a set of data points or measurements that have been recorded over time. In scientific experiments, this can refer to a graph or chart displaying the relationship between variables.

2. How do you interpret a trace?

To interpret a trace, you must first understand the variables being measured and the units used. Then, you can analyze the shape of the trace and any patterns or trends that may be present. It is also important to consider any outliers or errors in the data.

3. Can you provide an example of a trace?

One example of a trace could be a graph showing the change in temperature over time. The x-axis would represent time and the y-axis would represent temperature. The trace would show the trend of how the temperature changes over a given period of time.

4. How is a trace different from a trendline?

A trace refers to actual data points that have been measured and recorded, while a trendline is a line drawn to show the overall trend in the data. A trendline can be used to make predictions about future data points, while a trace shows the actual recorded values.

5. Can you show me how to create a trace in a scientific experiment?

To create a trace in a scientific experiment, you would need to collect data by taking measurements at specific intervals or time points. Then, you can plot the data points on a graph to create a trace. This can be done using software or by hand on graph paper.

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