In this paper

we gave a short proof of the singular value decomposition for the quaternion

matrices and proved:Theorem 2 Let (?) be the quaternion field

A∈(?)mxm then A =σ_1U_1+σ_2U 2 +…+σk Uk

where σ1 σ2

…

σk are all different non-nil singular values of A

and U1=σt-1 A(I(?)-(A~*A-σt~2 In)((A~*A-σt~2 In)2) (A* A-σt~2

In))

t=1

…

k.Theorem 3 Let A1

…

A?∈(?)mxm

then the following properties are equivalent:(i) There exist U∈Umxm

V∈Umxm such that all of UA?V* are diagonal with nonnegative diagonal entries

i=1

…t;(ii) All the matrices A

Aq~* and Ap~*A? are the positive semidefinite self-conjugate matrices of quaternions

p

q=1

…

t;(iii) There exist an ordering σp1

…

~σpmin(m

n) of the singular values of Ap

p=1

…

t

such that for all nonnegative real numbers x1 …

x

the singular values of are q=1

2

…

min(m

n).