Editing Diagonal matrix (section)

== Matrix operations ==
The operations of matrix addition and [[matrix multiplication]] are especially simple for diagonal matrices. Write {{math|diag(''a''<sub>1</sub>, ..., ''a<sub>n</sub>'')}} for a diagonal matrix whose diagonal entries starting in the upper left corner are {{math|''a''<sub>1</sub>, ..., ''a<sub>n</sub>''}}. Then, for [[matrix addition|addition]], we have

<math display=block>
  \operatorname{diag}(a_1,\, \ldots,\, a_n) + \operatorname{diag}(b_1,\, \ldots,\, b_n) = \operatorname{diag}(a_1 + b_1,\, \ldots,\, a_n + b_n)</math>

and for [[matrix multiplication]],

<math display=block>\operatorname{diag}(a_1,\, \ldots,\, a_n) \operatorname{diag}(b_1,\, \ldots,\, b_n) = \operatorname{diag}(a_1 b_1,\, \ldots,\, a_n b_n).</math>

The diagonal matrix {{math|diag(''a''<sub>1</sub>, ..., ''a<sub>n</sub>'')}} is [[invertible matrix|invertible]] [[if and only if]] the entries {{math|''a''<sub>1</sub>, ..., ''a<sub>n</sub>''}} are all nonzero. In this case, we have

<math display=block>\operatorname{diag}(a_1,\, \ldots,\, a_n)^{-1} = \operatorname{diag}(a_1^{-1},\, \ldots,\, a_n^{-1}).</math>

In particular, the diagonal matrices form a [[subring]] of the ring of all {{mvar|n}}-by-{{mvar|n}} matrices.

Multiplying an {{mvar|n}}-by-{{mvar|n}} matrix {{math|'''A'''}} from the ''left'' with {{math|diag(''a''<sub>1</sub>, ..., ''a<sub>n</sub>'')}} amounts to multiplying the {{mvar|i}}-th ''row'' of {{math|'''A'''}} by {{mvar|a<sub>i</sub>}} for all {{mvar|i}}; multiplying the matrix {{math|'''A'''}} from the ''right'' with {{math|diag(''a''<sub>1</sub>, ..., ''a<sub>n</sub>'')}} amounts to multiplying the {{mvar|i}}-th ''column'' of {{math|'''A'''}} by {{mvar|a<sub>i</sub>}} for all {{mvar|i}}.