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In linear algebra, a diagonal matrix is a matrix such that only elements on the diagonal are non-zero. The diagonal starts on the upper-left element and goes to the lower-right. If the matrix is non-square, the diagonal still starts at the upper-left.

Sometimes you'll see the following notation for specifying diagonal matricies:

D = Diagm×n(d1, d2, ..., dp)
This means that D is a m×n diagonal matrix containing the diagonal elements d1, d2, ..., dp from upper-left to lower-right. Here, p = min(m, n).

The rank of a diagonal matrix is the number of non-zero diagonal elements. The determinant of a square matrix is the product of the elements on the diagonal. Hence, a diagonal matrix has an inverse if and only if all the elements on the diagonal are non-zero. In that case, the inverse is given by:

D -1 = Diagn×m(1/d1, 1/d2, ..., 1/dp)
D -1 is a left inverse if mn. It's a right inverse if nm.

A square diagonal matrix is trivially symmetric. The only diagonal matrix which is also an orthogonal matrix is the identity matrix.

Geometrically, multiplication of a vector by a diagonal matrix can be interpreted as a non-uniform scaling of the elements of the vector.

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