Unlocking The Properties Of Diagonal Matrices In Linear Algebra: Definition, Operations And Applications

Diagonal matrix

A matrix having nonzero elements only in the diagonal running from the upper left to the lower right.

A diagonal matrix is a type of square matrix where all elements outside the main diagonal are zero. The main diagonal of a diagonal matrix contains the elements of the matrix. Diagonal matrices are often denoted by the symbol D, where the elements of the main diagonal are represented as d1, d2, d3, …, dn.

For example, a 3×3 diagonal matrix with elements 3, 2, and 1 on the main diagonal would be represented as:

| 3 0 0 |
| 0 2 0 |
| 0 0 1 |

Properties of Diagonal Matrices:
1. The inverse of a diagonal matrix exists if and only if all diagonal elements are non-zero. If a diagonal matrix has non-zero diagonal elements, the inverse matrix is also a diagonal matrix with the reciprocal of each diagonal element as the new diagonal element.
2. The determinant of a diagonal matrix is the product of its diagonal elements.
3. The trace of a diagonal matrix is the sum of its diagonal elements.

Diagonal matrices are commonly used in linear algebra and matrix operations because they are easy to work with and can simplify certain calculations. For example, multiplying a diagonal matrix by a vector or another diagonal matrix can be done efficiently by simply multiplying the corresponding diagonal elements.

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