Linear and multilinear algebra; matrix theory Definition

Diagonal matrix

Definition

Let $R$ be a ring and let $n \ge 0$ be an integer. A square matrix $D = (d_{ij})$ of order $n$ over $R$ is diagonal when $d_{ij} = 0$ for all $i \ne j$ . For $d_1, \dots, d_n \in R$ , the matrix $\operatorname{diag}(d_1, \dots, d_n)$ is the diagonal matrix with $d_{ii} = d_i$ for each $i$ .

Remarks

The product of two diagonal matrices is diagonal: $\operatorname{diag}(a_1, \dots, a_n) \operatorname{diag}(b_1, \dots, b_n) = \operatorname{diag}(a_1 b_1, \dots, a_n b_n)$ . Over a commutative ring, diagonal matrices commute with each other. The determinant of a diagonal matrix is the product of its diagonal entries. A matrix that is similar to a diagonal matrix is diagonalisable.

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Matrix definition
Square matrix definition
Diagonal matrix definition

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Tag	`0Q9PGXD9E4`
URI	`m2://0Q9PGXD9E4`
Kind	definition
Writing	W1 stated
MSC	15A99
Defines	Diagonal matrix
Uses	Square matrix

Diagonal matrix

Definition

Remarks

Reading path

Where this leads

Further reading

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