## What is the derivative of matrix determinant?

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In matrix calculus, Jacobi’s formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. where tr(X) is the trace of the matrix X. As a special case, It is named after the mathematician Carl Gustav Jacob Jacobi.

### How do you derive the covariance matrix?

Here’s how.

- Transform the raw scores from matrix X into deviation scores for matrix x. x = X – 11’X ( 1 / n )
- Compute x’x, the k x k deviation sums of squares and cross products matrix for x.
- Then, divide each term in the deviation sums of squares and cross product matrix by n to create the variance-covariance matrix.

**What is the differential of a matrix?**

A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to their derivatives.

**Is determinant differentiable?**

The determinant of a square matrix is a polynomial of its entries so it is infinitely differentiable.

## What is the Jacobian determinant?

The Jacobian determinant is used when making a change of variables when evaluating a multiple integral of a function over a region within its domain. To accommodate for the change of coordinates the magnitude of the Jacobian determinant arises as a multiplicative factor within the integral.

### What is the size of the covariance matrix of 3 dimensional data?

The 3D covariance, which is of size N3 × N3 where N3 is the size of the volume, is estimated from all the 2D covariances (of size N2 × N2) at different angles (θ, φ, ψ).

**What is the formula for calculating covariance?**

To calculate covariance, you can use the formula:

- Cov(X, Y) = Σ(Xi-µ)(Yj-v) / n.
- 6,911.45 + 25.95 + 1,180.85 + 28.35 + 906.95 + 9,837.45 = 18,891.
- Cov(X, Y) = 18,891 / 6.

**Can a covariant derivative be defined without a metric?**

It was soon noted by other mathematicians, prominent among these being Hermann Weyl, Jan Arnoldus Schouten, and Élie Cartan, that a covariant derivative could be defined abstractly without the presence of a metric.

## What is the covariant derivative of the basis vectors?

The covariant derivative of the basis vectors (the Christoffel symbols) serve to express this change. In a curved space, such as the surface of the Earth (regarded as a sphere), the translation is not well defined and its analog, parallel transport, depends on the path along which the vector is translated.

### Why does the covariant derivative change under a change in coordinates?

The covariant derivative is required to transform, under a change in coordinates, in the same way as a basis does: the covariant derivative must change by a covariant transformation (hence the name).

**What is the covariant derivative of a cylinder?**

The covariant derivative component is the component parallel to the cylinder’s surface, and is the same as that before you rolled the sheet into a cylinder. A covariant derivative is a (Koszul) connection on the tangent bundle and other tensor bundles: it differentiates vector fields in a way analogous to the usual differential on functions.