Primitive Matrix: Is M^k Positive?

You mention that you've tested this with MATLAB and it confirms this. You also ask if there is a theorem you can use to prove this. In summary, M is a product of n matrices, each with dimension 2n by 2n and a positive 3 by 3 block in the bottom right corner, and it is also a positive matrix. You may be able to use this information to prove that M is primitive.
  • #1
bob j
22
0
Hi All,
I have the following matrix

M = M_1 M_2 M_3 ... M_n

M is then a product of n matrices. Each of those has dimension 2n by 2n and has the same "look". Consider M_n: this matrix is equal to the identity matrix, 2n by 2n. The only thing different from the identity is that the 3 by 3 block on the bottom right of the matrix is composed of positive elements, and less than one. the other matrices are the same: matrix M_i has the 3 by 3 positive block placed at position 2(i-1) + 1, while the rest of the matrix is equal to the identity matrix.

My question is this. Is M primitive? Or, in other words, is M such that, for some k, M^k is positive? I tried with MATLAB and M is positive. Does anybody know if there is a theorem I can use?

Thanks
 
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  • #2
If I'm reading your post correctly, it seems to me that M is a product of positive matrices, and hence is a positive matrix itself.
 

FAQ: Primitive Matrix: Is M^k Positive?

What is a primitive matrix?

A primitive matrix is a square matrix with non-negative entries that can be raised to a power (k) to produce positive entries for all elements. It is a type of non-negative matrix that has special properties and is often used in linear algebra and graph theory.

How is a primitive matrix different from a regular matrix?

A primitive matrix must have all non-negative entries, whereas a regular matrix can have negative entries. Additionally, a primitive matrix must have a positive kth power, while a regular matrix may not have this property.

What does it mean for a matrix to be raised to a power?

When a matrix is raised to a power, it means the matrix is multiplied by itself a certain number of times. For example, M^2 would be equivalent to M multiplied by M, and M^3 would be equivalent to M multiplied by M multiplied by M.

Why is it important for M^k to be positive for a primitive matrix?

If M^k is positive for a primitive matrix, it means that the matrix has a special property known as "strong connectivity." This property is important in graph theory and can also be used to solve certain types of equations.

How can I determine if a matrix is primitive?

To determine if a matrix is primitive, you can use the Perron-Frobenius theorem, which states that a non-negative matrix with a positive kth power is primitive. Additionally, you can check if the matrix has strong connectivity by computing its eigenvalues and checking if they are all positive.

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