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PD Hoang
PD Hoang
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Another viewpoint, some inequalities of definite positive matrix, for example arithmetic-geometric means inequality, harmonic-geometric mean inquality,... are some special cases of more general things. That is called connection. Let $M_n^+ = \{ A \in M_n | A\ \text{is a semi-definite matrix} \}$

An connection is a binary operation $\sigma: M^+_n \times M^+_n \rightarrow M^+_n$ sastisfies:

1, $A \le B, C \le D \Rightarrow A \sigma C \le B \sigma D$.

2, $C(A \sigma B) C \le (CAC) \sigma (CBC)$.

3, $\sigma$ is continuous.

Mean is a connection with $1 \sigma 1 = 1$.

Example: $A \nabla B = \dfrac{A+B}{2}$, geometric mean is a connection.

We can get many ralations between the connection and some matric inequalities, for eaxample trace inequalities: http://en.wikipedia.org/wiki/Trace_inequalities

You can search on internet with keywords connections matrix inequalities.

Another viewpoint, some inequalities of definite positive matrix, for example arithmetic-geometric means inequality, harmonic-geometric mean inquality,... are some special cases of more general things. That is called connection. Let $M_n^+ = \{ A \in M_n | A\ \text{is a semi-definite matrix} \}$

An connection is a binary operation $\sigma: M^+_n \times M^+_n \rightarrow M^+_n$ sastisfies:

1, $A \le B, C \le D \Rightarrow A \sigma C \le B \sigma D$.

2, $C(A \sigma B) C \le (CAC) \sigma (CBC)$.

3, $\sigma$ is continuous.

Example: $A \nabla B = \dfrac{A+B}{2}$, geometric mean is a connection.

We can get many ralations between the connection and some matric inequalities, for eaxample trace inequalities: http://en.wikipedia.org/wiki/Trace_inequalities

You can search on internet with keywords connections matrix inequalities.

Another viewpoint, some inequalities of definite positive matrix, for example arithmetic-geometric means inequality, harmonic-geometric mean inquality,... are some special cases of more general things. That is called connection. Let $M_n^+ = \{ A \in M_n | A\ \text{is a semi-definite matrix} \}$

An connection is a binary operation $\sigma: M^+_n \times M^+_n \rightarrow M^+_n$ sastisfies:

1, $A \le B, C \le D \Rightarrow A \sigma C \le B \sigma D$.

2, $C(A \sigma B) C \le (CAC) \sigma (CBC)$.

3, $\sigma$ is continuous.

Mean is a connection with $1 \sigma 1 = 1$.

Example: $A \nabla B = \dfrac{A+B}{2}$, geometric mean is a connection.

We can get many ralations between the connection and some matric inequalities, for eaxample trace inequalities: http://en.wikipedia.org/wiki/Trace_inequalities

You can search on internet with keywords connections matrix inequalities.

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PD Hoang
PD Hoang
  • 19
  • 4

Another viewpoint, some inequalities of definite positive matrix, for example arithmetic-geometric means inequality, harmonic-geometric mean inquality,... are some speacialspecial cases of more general things. That is called connection. Let $M_n^+ = \{ A \in M_n | A\ \text{is a semi-definite matrix} \}$

An connection is a binary operation $\sigma: M^+_n \times M^+_n \rightarrow M^+_n$ sastisfies:

1, $A \le B, C \le D \Rightarrow A \sigma C \le B \sigma D$.

2, $C(A \sigma B) C \le (CAC) \sigma (CBC)$.

3, $\sigma$ is continuous.

Example: $A \nabla B = \dfrac{A+B}{2}$, geometric mean is a connection.

We can get many ralations between the connection and some matric inequalities, for eaxample trace inequalities: http://en.wikipedia.org/wiki/Trace_inequalities

You can search on internet with keywords connections matrix inequalities.

Another viewpoint, some inequalities of definite positive matrix, for example arithmetic-geometric means inequality, harmonic-geometric mean inquality,... are some speacial cases of more general things. That is called connection. Let $M_n^+ = \{ A \in M_n | A\ \text{is a semi-definite matrix} \}$

An connection is a binary operation $\sigma: M^+_n \times M^+_n \rightarrow M^+_n$ sastisfies:

1, $A \le B, C \le D \Rightarrow A \sigma C \le B \sigma D$.

2, $C(A \sigma B) C \le (CAC) \sigma (CBC)$.

3, $\sigma$ is continuous.

Example: $A \nabla B = \dfrac{A+B}{2}$, geometric mean is a connection.

We can get many ralations between the connection and some matric inequalities, for eaxample trace inequalities: http://en.wikipedia.org/wiki/Trace_inequalities

You can search on internet with keywords connections matrix inequalities.

Another viewpoint, some inequalities of definite positive matrix, for example arithmetic-geometric means inequality, harmonic-geometric mean inquality,... are some special cases of more general things. That is called connection. Let $M_n^+ = \{ A \in M_n | A\ \text{is a semi-definite matrix} \}$

An connection is a binary operation $\sigma: M^+_n \times M^+_n \rightarrow M^+_n$ sastisfies:

1, $A \le B, C \le D \Rightarrow A \sigma C \le B \sigma D$.

2, $C(A \sigma B) C \le (CAC) \sigma (CBC)$.

3, $\sigma$ is continuous.

Example: $A \nabla B = \dfrac{A+B}{2}$, geometric mean is a connection.

We can get many ralations between the connection and some matric inequalities, for eaxample trace inequalities: http://en.wikipedia.org/wiki/Trace_inequalities

You can search on internet with keywords connections matrix inequalities.

Source Link
PD Hoang
PD Hoang
  • 19
  • 4

Another viewpoint, some inequalities of definite positive matrix, for example arithmetic-geometric means inequality, harmonic-geometric mean inquality,... are some speacial cases of more general things. That is called connection. Let $M_n^+ = \{ A \in M_n | A\ \text{is a semi-definite matrix} \}$

An connection is a binary operation $\sigma: M^+_n \times M^+_n \rightarrow M^+_n$ sastisfies:

1, $A \le B, C \le D \Rightarrow A \sigma C \le B \sigma D$.

2, $C(A \sigma B) C \le (CAC) \sigma (CBC)$.

3, $\sigma$ is continuous.

Example: $A \nabla B = \dfrac{A+B}{2}$, geometric mean is a connection.

We can get many ralations between the connection and some matric inequalities, for eaxample trace inequalities: http://en.wikipedia.org/wiki/Trace_inequalities

You can search on internet with keywords connections matrix inequalities.