Proof of ##M^n## (matrix multiplication problem)

In summary: In other words, the order in the proof does not matter.Homework Statement: Please see belowRelevant Equations: Please see below
  • #1
ChiralSuperfields
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Homework Statement
Please see below
Relevant Equations
Please see below
For,
1683666942440.png

Does anybody please know why they did not change the order in the second line of the proof? For example, why did they not rearrange the order to be ##M^n = (DP^{-1}P)(DP^{-1}P)(DP^{-1}P)(DP^{-1}P)---(DP^{-1}P)## for to get ##M^n = (DI)(DI)(DI)(DI)---(DI) = D^n##

Many thanks!
 
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  • #2
ChiralSuperfields said:
Homework Statement: Please see below
Relevant Equations: Please see below

For,
View attachment 326261
Does anybody please know why they did not change the order in the second line of the proof? For example, why did they not rearrange the order to be ##M^n = (DP^{-1}P)(DP^{-1}P)(DP^{-1}P)(DP^{-1}P)---(DP^{-1}P)## for to get ##M^n = (DI)(DI)(DI)(DI)---(DI) = D^n##

Many thanks!
They did exactly what you proposed only with one more step that shows how the associativity law is necessary here. We first need ##(AB)B^{-1}=A(BB^{-1})=AI=A## before we are allowed to write ##A##. It is important to recognize which laws of matrix multiplication are actually used. Here it's the law of associativity ##(AB)B^{-1}=A(BB^{-1})##, the definition of an inverse ##BB^{-1}=I## and the definition of the neutral element ##AI=A.## So every single property of matrix multiplication, except for the distributive law since there is no addition here, has actually been used.
 
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  • #3
ChiralSuperfields said:
Homework Statement: Please see below
Relevant Equations: Please see below

For,
View attachment 326261
Does anybody please know why they did not change the order in the second line of the proof? For example, why did they not rearrange the order to be ##M^n = (DP^{-1}P)(DP^{-1}P)(DP^{-1}P)(DP^{-1}P)---(DP^{-1}P)## for to get ##M^n = (DI)(DI)(DI)(DI)---(DI) = D^n##

Many thanks!
Because that would not be correct. Matrix multiplication is not commutative, and it is definitely not true that a matrix, ##M^n## that is probably non-diagonal, is equal to the diagonal matrix ##D^n##.
 
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  • #4
@FactChecker has a good point here. I didn't see that you switched the order of ##P## and ##D.##

The only matrices that can be swapped are the diagonal matrices with the same value on the entire diagonal.
$$
A\cdot B = B\cdot A \text{ for all matrices } A \Longleftrightarrow D=\operatorname{diag}(d,d,\ldots,d)
$$

So if we do not have any specific information about ##A,## we must treat it like an arbitrary matrix. And that leaves us with ##\begin{pmatrix}d&0&\ldots&0\\0&d&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&d\\ \end{pmatrix}## as the only matrix that commutes with ##A.## If we consider a specific matrix ##A,## then there are possibly more matrices that commute with ##A##. (Commute means ##A\cdot B=B\cdot A.##) However, if all these matrices are as before, then ##PD\neq DP.##
 
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1. What is "Proof of ##M^n##"?

"Proof of ##M^n##" refers to the mathematical process of proving the validity of a statement or equation involving the multiplication of matrices. This involves using logical reasoning and mathematical principles to show that the statement is true for all possible values of the matrices involved.

2. Why is "Proof of ##M^n##" important?

Understanding the proof of ##M^n## is important for several reasons. Firstly, it allows us to verify the accuracy of mathematical statements involving matrix multiplication. Additionally, it helps us to gain a deeper understanding of the properties and behaviors of matrices, which can be applied in various fields such as engineering, physics, and computer science.

3. What are the key steps in proving ##M^n##?

The key steps in proving ##M^n## involve defining the matrices involved, setting up the equation or statement to be proved, and then using mathematical techniques such as algebra, induction, or direct proof to show that the statement is true for all values of the matrices. It is also important to clearly state any assumptions or properties of matrices that are being used in the proof.

4. Can you give an example of a "Proof of ##M^n##"?

One example of a "Proof of ##M^n##" is the proof of the associative property of matrix multiplication. This proof involves setting up an equation with three matrices and then using the properties of matrix multiplication to show that the order of multiplication does not affect the final result.

5. Are there any common mistakes to avoid in "Proof of ##M^n##"?

Yes, there are some common mistakes to avoid in "Proof of ##M^n##". One mistake is assuming that the commutative property holds for matrix multiplication, as this is not always the case. Another mistake is not clearly stating all assumptions and properties used in the proof. It is also important to check for errors in calculations and to provide clear and logical explanations for each step in the proof.

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