| Summary: | Let <i>A</i> be an <i>n</i>-by-<i>n</i> matrix. The numerical range of <i>A</i> is defined as <inline-formula> <math display="inline"> <semantics> <mrow> <mi>W</mi> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <msup> <mi>x</mi> <mo>*</mo> </msup> <mi>A</mi> <mi>x</mi> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> <mo>,</mo> <msup> <mi>x</mi> <mo>*</mo> </msup> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>}</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. The Moore–Penrose inverse <inline-formula> <math display="inline"> <semantics> <msup> <mi>A</mi> <mo>+</mo> </msup> </semantics> </math> </inline-formula> of <i>A</i> is the unique matrix satisfying <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <msup> <mi>A</mi> <mo>+</mo> </msup> <mi>A</mi> <mo>=</mo> <mi>A</mi> <mo>,</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mi>A</mi> <msup> <mi>A</mi> <mo>+</mo> </msup> <mo>=</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mo>,</mo> <msup> <mrow> <mo>(</mo> <mi>A</mi> <msup> <mi>A</mi> <mo>+</mo> </msup> <mo>)</mo> </mrow> <mo>*</mo> </msup> <mo>=</mo> <mi>A</mi> <msup> <mi>A</mi> <mo>+</mo> </msup> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mi>A</mi> <mo>)</mo> </mrow> <mo>*</mo> </msup> <mo>=</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mi>A</mi> </mrow> </semantics> </math> </inline-formula>. This paper investigates the numerical range of the Moore–Penrose inverse <inline-formula> <math display="inline"> <semantics> <msup> <mi>A</mi> <mo>+</mo> </msup> </semantics> </math> </inline-formula> of a matrix <i>A</i>, and examines the relation between the numerical ranges <inline-formula> <math display="inline"> <semantics> <mrow> <mi>W</mi> <mo>(</mo> <msup> <mi>A</mi> <mo>+</mo> </msup> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>W</mi> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>.
|
|---|