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Problem

Consider a generic matrix $A$, we are going to think of a simple case by taking into consideration a $3 \times 3$ matrix:

$$ A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3}\\ a_{2,1} & a_{2,2} & a_{2,3}\\ a_{3,1} & a_{3,2} & a_{3,3}\\ \end{pmatrix} $$

Consider now having $A'$ as:

$$ A' = \begin{pmatrix} a'_{1,1} & a'_{1,2} & a'_{1,3}\\ a'_{2,1} & a'_{2,2} & a'_{2,3}\\ a'_{3,1} & a'_{3,2} & a'_{3,3}\\ \end{pmatrix} $$

The following holds:

$$a'_{i,j} \leq a_{i,j}$$

Question

I would like to know if the following:

$$|A'| \leq |A|$$

If it holds, can you prove it?

Another problem

What if we considered:

$$ a_{i,j} \leq 1, a'_{i,j} \leq 1 $$

Considering also that $A$ is a stochastic matrix?

This does not mean that both $A$ and $A'$ are stochastic. I am considering $A$ stochastic and $A'$ obtained as a reduced version of $A$ so that $A'$ is not stochastic but its values are all between 0 and 1.

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1  
In general, the determinant of an $n\times n$ matrix is a rather complicated polynomial in the $a_{ij}$, and certainly there's no reason to expect this to be increasing with respect to all variables. –  Aaron Mazel-Gee May 20 '12 at 6:47

2 Answers 2

up vote 11 down vote accepted

The claim is false. For instance $$A' = \begin{pmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1 \end{pmatrix}$$ and $$A = \begin{pmatrix}2 & 2 & 2\\2 & 2 & 2\\2 & 2 & 2 \end{pmatrix}$$ Clearly, $A'_{ij} \leq A_{ij}$, whereas $$\det(A') = 1 > 0 \det(A)$$

EDIT

If $A$ and $A'$ are both stochastic matrices, then $A'_{ij} \leq A_{ij}$ gives us $A'_{ij} = A_{ij}$ since $$1 = \displaystyle \sum_{j=1}^{3} A'_{ij} \leq \sum_{j=1}^{3} A_{ij} = 1, \,\forall i \in \{1,2,3\}$$

EDIT

If $A$ is stochastic, but $A'$ is not stochastic, then again it is false. For instance, $$A = \begin{pmatrix}\frac13 & \frac13 & \frac13\\\frac13 & \frac13 & \frac13\\\frac13 & \frac13 & \frac13 \end{pmatrix}$$ and $$A' = \begin{pmatrix}\frac13 & 0 & 0\\0 & \frac13 & 0\\0 & 0 & \frac13 \end{pmatrix}$$

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It looks like we came up with the same example:-) –  algebra_fan May 20 '12 at 6:46
    
@algebra_fan :) Interesting. –  user17762 May 20 '12 at 6:46
    
Thank you :) What if A were stochastic? –  Andry May 20 '12 at 6:53
    
Please read more carefully, now I pointed it better but before I stated that A only is stochastic. –  Andry May 20 '12 at 7:21

This is false. Let $A$ be the matrix all of whose entries are equal to $2$ and $A'$ be the identity matrix. Note that $\det(A) = 0$ since its columns are linearly dependent, while $\det(A') = 1$.

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Thank you very much, please see edits. –  Andry May 20 '12 at 6:53
    
If $A$ and $A'$ are stochastic, then $A = A'$ as Marvis showed in the response above. –  algebra_fan May 20 '12 at 7:07
    
No I said that only A is stochastic :P –  Andry May 20 '12 at 7:27

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