# Finding satisfying matrix for a particular condition

• Seydlitz
In summary, the conversation revolves around finding a real square matrix that satisfies the condition AA^T = -I. It is mentioned that an orthogonal matrix does not work, and only even square matrices with real determinants can satisfy the condition. Trivial and symmetric matrices also do not work. It is concluded that there is no such matrix because AA^T is always positive semidefinite while -I is negative definite.
Seydlitz
Hello guys,

I'm trying rather hard to find a real square matrix that will satisfy this ##AA^T=-I##. The first thing that comes to my mind is of course orthogonal matrix. But clearly it isn't. In fact, ##A^T=-A^{-1}## in order for it to work. The condition is pretty strict if you also consider that only even square matrix can satisfy this condition for real determinant and hence real matrix. (The reason is ##-I## will have a determinant value of ##-1## if the matrix is odd.)

Trivial matrix and symmetric matrix will not even work even though ##I(-I)=-I## because ##-I## is not the transpose of ##I##.

Moreover, I'm not sure if square matrix will work because clearly the diagonal of ##AA^T## cannot be ##-1## if ##A## is real matrix.

##iI## works, but then, it's not real.

There is no such matrix. The reason is that ##AA^T## is positive semidefinite, whereas ##-I## is negative definite.

To see this, let ##x## be any nonzero real vector of the appropriate dimension. Then ##x^T(AA^T)x = (x^TA)(A^Tx) = (A^T x)^T (A^Tx) \geq 0##. On the other hand, ##x^T (-I) x = -(x^T x) < 0##.

Thanks jbunnii for your explanation. I was already quite convinced that it is, seeing the case using 2x2 matrix.

## 1. How do you define a satisfying matrix?

A satisfying matrix is a square matrix where the values of the main diagonal are equal to 1, and all other values are either 0 or 1. This means that the matrix is symmetric and has only two distinct values.

## 2. What is the importance of finding a satisfying matrix?

A satisfying matrix is important in a variety of applications, such as game theory, computer science, and statistics. It is also used in linear algebra to represent binary relationships between different elements.

## 3. What is the process for finding a satisfying matrix for a particular condition?

The process for finding a satisfying matrix for a particular condition involves first determining the size of the matrix, then filling in the main diagonal with 1's. Next, the remaining values can be filled in following certain rules or conditions, depending on the specific requirements of the problem.

## 4. Are there any limitations to finding a satisfying matrix?

Yes, there are limitations to finding a satisfying matrix. It is only possible to find a satisfying matrix if the size of the matrix is equal to the number of possible combinations of 0 and 1 in a symmetric matrix. This means that larger matrices may not have a satisfying solution.

## 5. Can a satisfying matrix be used for any type of problem?

No, a satisfying matrix is specifically designed to represent binary relationships and may not be applicable to all types of problems. It is most commonly used in situations where only two outcomes are possible, such as in decision-making processes or in representing connections in a network.

Replies
14
Views
2K
Replies
1
Views
1K
Replies
8
Views
2K
Replies
34
Views
2K
Replies
2
Views
2K
Replies
1
Views
1K
Replies
1
Views
2K
Replies
3
Views
1K
Replies
1
Views
889
Replies
2
Views
1K