Kernel (matrix) - Definition

Definition

The kernel of an m × n matrix A with coefficients in a field K (typically the field of the real numbers or of the complex numbers) is the set

where 0 denotes the zero vector with m components. The matrix equation Ax = 0 is equivalent to a homogeneous system of linear equations:

$\mathbf{A}\textbf{x}=\textbf{0} \;\;\Leftrightarrow\;\; \begin{alignat}{6} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = 0& \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = 0& \\ \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; && \vdots\,& \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = 0. & \end{alignat}$

From this viewpoint, the null space of A is the same as the solution set to the homogeneous system.

Read more about this topic: Kernel (matrix)

Famous quotes containing the word definition:

“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)

“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)