2025-06-08

1152: Positive-Definite Hermitian Matrix Can Be Transformed to Identity by Unitary Matrix Multiplied by Positive Diagonal Matrix from Right

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description/proof of that positive-definite Hermitian matrix can be transformed to identity by unitary matrix multiplied by positive diagonal matrix from right

Topics


About: matrix

The table of contents of this article


Starting Context



Target Context


  • The reader will have a description and a proof of the proposition that any positive-definite Hermitian matrix can be transformed to the identity by a unitary matrix multiplied by a positive diagonal matrix from right.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
M: { the positive-definite Hermitian matrices }
//

Statements:
U{ the unitary matrices },D{ the positive diagonal matrices }((UD)M(UD)=I)
//

(UD)M(UD)=DUMUD=DtU1MUD, because U is unitary and D is real. =DU1MUD, because D is symmetric.

When M is real symmetric, which is Hermitian, U can be taken to be orthogonal, which is unitary, and (UD)M(UD)=(UD)tM(UD)=I.


2: Proof


Whole Strategy: Step 1: see that there is a U such that M=UMU=(λ10...00λ2...0...0...0λn) is a positive diagonal matrix; Step 2: take D=(1/λ10...001/λ2...0...0...01/λn), and see that (UD)M(UD)=I.

Step 1:

There is a U such that M=UMU=(λ10...00λ2...0...0...0λn) is a real diagonal matrix, by the proposition that any Hermitian matrix can be diagonalized by a unitary matrix to be inevitably real.

As M is positive-definite, each λj is positive: otherwise, v(UMU)v would be non-positive for a nonzero v and =(Uv)M(Uv) with Uv nonzero with non-positive result, a contradiction: Uv0 because (Uv)(Uv)=vUUv=vIv=vv0.

Step 2:

Let us define D:=(1/λ10...001/λ2...0...0...01/λn).

(UD)M(UD)=DUMUD=D(UMU)D=(1/λ10...001/λ2...0...0...01/λn)(λ10...00λ2...0...0...0λn)(1/λ10...001/λ2...0...0...01/λn)=I.


References


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