# The Forward Direction of "V: a + iTa=b+iTb, iff a=b"

• CoachZ
In summary, the conversation discusses proving the equivalence of a + iTa = b + iTb and a = b in a complex inner product space with a self-adjoint linear operator. The forward direction of the proof is causing difficulty, but the concept of real eigenvalues is mentioned as a possible approach. The conversation ends with the acknowledgement that the concept of non-singular matrices may also be helpful in solving the problem.
CoachZ
Let V be a complex inner product space and T a self-adjoint linear operator on V.

I'm trying to show a + iTa = b + iTb, iff a = b. The converse is trivial. The forward direction is getting me for some reason. Perhaps it's too late on a Friday night that my mind is completely gone. Any suggestions...

CoachZ said:
Let V be a complex inner product space and T a self-adjoint linear operator on V.

I'm trying to show a + iTa = b + iTb, iff a = b. The converse is trivial. The forward direction is getting me for some reason. Perhaps it's too late on a Friday night that my mind is completely gone. Any suggestions...
So a- b= -iT(a- b) and T(a- b)= i(a- b). Either a- b= 0 or i is an eigenvalue of T. But all eigenvalues of self-adjoint operators are real.

Haha, I woke up this morning thinking about I + iT and I - iT, and how they are both non-singular, which uses the theorem of real eigenvalues, and I was thinking that a + iTa = b + iTb probably uses similar concepts and ideas to solve. Thanks for the help!

## What is "The Forward Direction of "V: a + iTa=b+iTb, iff a=b"?

"The Forward Direction of "V: a + iTa=b+iTb, iff a=b" is a mathematical statement that describes the relationship between two complex numbers, a and b, where a is equal to b when multiplied by the imaginary unit i.

## What is a complex number?

A complex number is a number that contains both a real part and an imaginary part, typically written in the form a+bi, where a and b are real numbers and i is the imaginary unit (√-1).

## What is the significance of the "Forward Direction" in this statement?

The "Forward Direction" refers to the implication that if a is equal to b when multiplied by i, then the complex numbers a+bi and b+bi are equal. This is a fundamental property of complex numbers and is used in many mathematical proofs.

## What does "iff" mean in this statement?

"Iff" stands for "if and only if" and indicates that both sides of the equation are equivalent and cannot be separated.

## How is this statement relevant in science?

This statement is relevant in science as it is used in many fields, such as physics and engineering, to describe and analyze complex systems and phenomena. It is also used in various mathematical models and equations to solve problems and make predictions.

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