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Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$

$$2AA^T+BB^T+CC^T=(4n+4)I-4J$$

where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix of ones. My questions:

1. Are all orders good? Here the term "good" means that the equation has at least one solution.

2. How to find $A$, $B$ and $C$ when $n$ is given?

Any comment/answer would be appreciated.

Related

• One question on circulant $\pm1$ matrices

Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$

$$2AA^T+BB^T+CC^T=(4n+4)I-4J$$

where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix of ones. My questions:

1. Are all orders good? Here the term "good" means that the equation has at least one solution.

2. How to find $A$, $B$ and $C$ when $n$ is given?

Any comment/answer would be appreciated.

Related

• One question on circulant $\pm1$ matrices

Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$

$$2AA^T+BB^T+CC^T=(4n+4)I-4J$$

where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix of ones. My questions:

1. Are all orders good? Here the term "good" means that the equation has at least one solution.

2. How to find $A$, $B$ and $C$ when $n$ is given?

Any comment/answer would be appreciated.

Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$

$$2AA^T+BB^T+CC^T=(4n+4)I-4J$$

where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix of ones. My questions:

1. Are all orders good? Here the term "good" means that the equation has at least one solution.

2. How to find $A$, $B$ and $C$ when $n$ is given?

Any comment/answer would be appreciated.

Related

• One question on circulant $\pm1$ matrices

Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$

$$2AA^T+BB^T+CC^T=(4n+4)I-4J$$

where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix of ones. My questions:

1. Are all orders good? Here the term "good" means that the equation has at least one solution.

2. How to find $A$, $B$ and $C$ when $n$ is given?

Any comment/answer would be appreciated.

Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$

$$2AA^T+BB^T+CC^T=(4n+4)I-4J$$

where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix of ones. My questions:

1. Are all orders good? Here the term "good" means that the equation has at least one solution.

2. How to find $A$, $B$ and $C$ when $n$ is given?

Any comment/answer would be appreciated.