Hermitian Conjugate Matrix

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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Do you know, or can you guess, the technical symbols? Well, let's see! Show Instructions

Codes for the ⊹ Symbol

The Symbol	⊹	copy
Alt Code	Alt 8889	copy
HTML Code	⊹	copy
HTML Entity	⊹	copy
CSS Code	\22B9	copy
Hex Code	⊹	copy
Unicode	U+22B9	copy

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: &#8889;</b>	My symbol: ⊹

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

HTML Text	Output
<b>My symbol: &hercon;</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: &#x22B9;</b>	My symbol: ⊹

On the assumption that you already have your canvas and the context set up, use the Hex code in the format 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.)

In JavaScript, the syntax is \uXXXX. So, our example would be \u22B9. (Note that the format is 4 hexadecimal characters.)

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