## The Mathematical Symbol "Hermitian Conjugate Matrix (⊹)"

Hermitian Conjugate Matrix in Mathematics

In the realm of mathematics, especially within linear algebra and quantum mechanics, the Hermitian conjugate of a matrix holds significant importance. This operation involves taking the transpose of the matrix and then taking the complex conjugate of each of its entries. The Hermitian conjugate of a matrix \( A \) is often denoted as \( A^† \) or \( A^* \).

## Definition

The Hermitian conjugate \( A^† \) of a matrix \( A \) is defined by:

\[ (A^†)_{ij} = \overline{(A_{ji})} \]where \( \overline{(A_{ji})} \) denotes the complex conjugate of the entry in the \( j \)-th row and \( i \)-th column of \( A \).

## Examples

**Example 1:**For a 2x2 matrix \[ A = \begin{bmatrix} 1 + i & 2 \\ 3 & 4 - i \end{bmatrix} \] The Hermitian conjugate \( A^† \) is: \[ A^† = \begin{bmatrix} 1 - i & 3 \\ 2 & 4 + i \end{bmatrix} \]**Example 2:**For a vector \[ v = \begin{bmatrix} 1 + i \\ 2 - i \end{bmatrix} \] The Hermitian conjugate \( v^† \) is: \[ v^† = [1 - i, 2 + i] \]

To conclude, the Hermitian conjugate is a crucial matrix operation, especially when dealing with complex matrices in fields like quantum mechanics and signal processing. It provides a standardized way to transpose and conjugate matrices, leading to many essential properties and applications in theoretical and applied mathematics.

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## Codes for the ⊹ Symbol

The Symbol | ⊹ | |

Alt Code | Alt 8889 | |

HTML Code | ⊹ | |

HTML Entity | ⊹ | |

CSS Code | \22B9 | |

Hex Code | ⊹ | |

Unicode | U+22B9 |

## How To Insert the ⊹ Symbol

(Method 1) Copy and paste the symbol.

The easiest way to get the ⊹ symbol is to copy and paste it into your document.Bear in mind that this is a UTF-8 encoded character. It must be encoded as UTF-8 at all stages (copying, replacing, editing, pasting), otherwise it will render as random characters or the dreaded �.

(Method 2) Use the "Alt Code."

If you have a keyboard with a numeric pad, you can use this method. Simply hold down the Alt key and type 8889. When you lift the Alt key, the symbol appears. ("Num Lock" must be on.)(Method 3) Use the HTML Decimal Code (for webpages).

HTML Text | Output |
---|---|

<b>My symbol: ⊹</b> | My symbol: ⊹ |

(Method 4) Use the HTML Entity Code (for webpages).

HTML Text | Output |
---|---|

<b>My symbol: ⊹</b> | My symbol: ⊹ |

(Method 5) Use the CSS Code (for webpages).

CSS and HTML Text | Output |
---|---|

<style> span:after { content: "\22B9";} </style> <span>My symbol:</span> | My symbol: ⊹ |

(Method 6) Use the HTML Hex Code (for webpages and HTML canvas).

HTML Text | Output |
---|---|

<b>My symbol: ⊹</b> | My symbol: ⊹ |

**0x22B9**to place the ⊹ symbol on your canvas. For example:

JavaScript Text |
---|

const x = "0x"+"E9" ctx.fillText(String.fromCodePoint(x), 5, 5); |

Output |

⊹ |

(Method 7) Use the Unicode (for various, e.g. Microsoft Office, JavaScript, Perl).

The Unicode for ⊹ is**U+22B9**. The important part is the hexadecimal number after the

**U+**, which is used in various formats. For example, in Microsoft Office applications (e.g. Word, PowerPoint), do the following:

Type | Output |
---|---|

22B9 [Hold down Alt] [Press x] | ⊹ (The 22B9 turns into ⊹. Note that you can omit any leading zeros.) |

JavaScript Text | Output |
---|---|

let str = "\u22B9" document.write("My symbol: " + str) | My symbol: ⊹ |