# Dab solver - Matrix calculus

./dab_solver.py -page:
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==  Derivatives with vectors  ==

{{Main|Vector calculus}}
Because vectors are matrices with only one column, the simplest matrix derivatives are vector derivatives.

The notations developed here can accommodate the usual operations of [[vector calculus]] by identifying the space M(n,1) of n-vectors with the [[Euclidean space]] Rn, and the scalar M(1,1) is identified with R. The corresponding concept from vector calculus is indicated at the end of each subsection.

NOTE: The discussion in this section assumes the [[#Layout conventions|numerator layout convention]] for pedagogical purposes.  Some authors use different conventions.  The section on [[#Layout conventions|]] discusses this issue in greater detail.  The identities given further down are presented in forms that can be used in conjunction with all common layout conventions.

===  Vector-by-scalar  ===

The [[derivative]] of a [[Euclidean vector|vector]]
$\mathbf{y} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_m \\ \end{bmatrix}$,
by a  x is written (in [[#Layout conventions|numerator layout notation]]) as

:$\frac{\partial \mathbf{y}}{\partial x} = \begin{bmatrix} \frac{\partial y_1}{\partial x}\\ \frac{\partial y_2}{\partial x}\\ \vdots\\ \frac{\partial y_m}{\partial x}\\ \end{bmatrix}.$

In [[vector calculus]] the derivative of a vector y with respect to a scalar x is known as the [[tangent vector]] of the vector y, $\frac{\partial \mathbf{y}}{\partial x}$. Notice here that y:R $\rightarrow$ Rm.

Example Simple examples of this include the [[velocity]] vector in [[Euclidean space]], which is the [[tangent vector]] of the [[Position (vector)|]] vector (considered as a function of time). Also, the [[acceleration]] is the tangent vector of the velocity.

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