O transposição de matriz is a fundamental operation in álgebra linear that involves flipping a matrix over its diagonal. This means that the element at row i and column j of the original matrix becomes the element at row j and column i in the transposed matrix. Mathematically, if A is a matrix, its transpose is denoted as AT or A‘.
Por exemplo, considere uma matriz:
A =
[
[1, 2, 3]
[4, 5, 6]
[7, 8, 9]
]
Sua transposta seria:
AT =
[
[1, 4, 7]
[2, 5, 8]
[3, 6, 9]
]
The transpose operation has several important applications in various fields, including gráficos computacionais, aprendizado de máquina, and dados útil. For instance, in computer graphics, transposing matrices is often necessary for transforming shapes and coordinates. In machine learning, transposed matrices are used in algorithms para calcular gradientes e otimizar modelos de forma eficiente.
In terms of properties, the transpose of a sum of matrices is equal to the sum of their transposes, and the transpose of a product of matrices is equal to the product of their transposes in reverse order: (AB)T = BTAT.