Help with Matrix PQ=QR - Find a Simple Answer

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In summary, There are multiple solutions to this problem, but some possible solutions include: 1) P and R are zero matrices with the same dimensions as Q, while Q is a null matrix. 2) P, Q, and R are all identity matrices with the same dimensions. 3) P and R are powers of the same matrix, while Q is a scalar matrix. Additionally, if R is similar to P, then there exists a regular matrix Q such that R = Q^-1 P Q and QR = PQ. Similar matrices can represent linear operators in different basis sets.
  • #1
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There must be a simple answer to this problem but ill be damed if i can find it.

I need to find matrix P,Q,R So that PQ=QR

Ive tried so many times but i can't solve it, I've been to three different applicable maths books looking for help but they all where dead ends.

Could someone please help me.
 
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  • #2
Which matrices if any are given?
 
  • #3
One solution is for P, Q, and R to each be a zero matrix with the appropriate dimensions.
 
  • #4
Well, if P,Q,R are the identity matrix then that would work, but at the moment it's not a well posed problem.
 
  • #5
1)P and R can be any matrix of the same order as that of Q and Q must be a null matrix.
2)All the three should be identity matrices of the same order.
 
  • #6
There're an infinate amount of solutions to this problem.
In addition to everything said above the equation will hold if all three matricies are powers of some matrix, or if they're scalar matricies.
 
  • #7
Assume Q is regular, and P, R are square matrices. If you multiply PQ = QR from the left with Q^-1, you obtain Q^-1 P Q = R, which implies that R is similar to P.

Further on, if R is similar to P, then there exists a regular matrix Q such that R = Q^-1 P Q. Multiply from the left with Q and obtain QR = PQ.

So, PQ = QR <=> R is similar to P. That could be one point of view. An example of similar matrices are matrix representations of linear operators in different basis sets.
 

What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is commonly used in mathematics, physics, and computer science to represent data and perform calculations.

What is a PQ=QR matrix?

A PQ=QR matrix is a matrix equation that represents the multiplication of two matrices, PQ and QR. In this equation, the number of columns in the first matrix (PQ) must equal the number of rows in the second matrix (QR).

How do I solve for a matrix PQ=QR?

To solve for a matrix PQ=QR, you can use the properties of matrix multiplication. First, determine the dimensions of the matrices PQ and QR. Then, multiply the corresponding elements in each row of PQ with each column of QR. Finally, combine the products to create the resulting matrix.

What is a simple answer to finding a matrix PQ=QR?

A simple answer to finding a matrix PQ=QR is to use the properties of matrix multiplication and follow the steps outlined in the previous question. It is important to ensure that the number of columns in PQ is equal to the number of rows in QR for the equation to be solvable.

What are some real-world applications of solving a matrix PQ=QR?

Matrix PQ=QR is commonly used in fields such as engineering, economics, and computer graphics to solve systems of linear equations, analyze data, and perform transformations. It is also used in machine learning and data science for tasks such as image and signal processing, recommendation systems, and natural language processing.

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