A

NATIONAL BUREAU OF STANDARDS REPORT

2093

CHARACTERISTIC ROOTS OF QUATERNION MATRICES

by

Olga Taussky

l • .

.

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT

w NBS REPORT

1 1 02-20-1 1 0^ December 3? 1952 2093

Characteristic Roots of Quaternion Matrices

by

Olga Taussky

National Bureau of Standards

The publication, repri

unless permission Is ol

25, D, C, Such perml

cally prepared If that

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

n part, Is prohibited

mdards, Washington

ort has been specif

ort for Its own use

CHARACTERISTIC ROOTS OF QUATERNION MATRICES

by Olga Tauesky

In two- resent publications [1]* [2] It was shown

that for matrices of (real) quaternion elements an eigenvalue theory can he developed similar to that for complex

numberBo If A is ouch a matrix then quaternion ©laments

ana quaternion vectore x can he found such that

Ax » x A o This note contains some remarks to supplement the

results obtained In [1] and [2], It is* however, selfcontained*

For complex'-matrices It is known that the transposed

matrix has the same roots, but different vectors . Let

_ matrix

A be a compleV and x a vector which corresponds to

one of its character! s t io roots A ; let^w be any root

of A1 and y a characteristic vector which corresponds to

it; then It is known that

A y

6 x a* Jk. y’x 0 From this one concludes that either A ® /or y«x = 0*

in the quaternion case the following generalization

holds

;

Theorem 1* Let A be a quaternion matrix* A one of Its

characteristic roots, x a corresponding vector* Let A®

be the transposed and conjugate matrix and u one of its

I'oots with y as a corresponding vector* Than

'

, The conjugate matrix had to he introduced since

the rule (AB)® » BI AI

. does not hold for quaternion

; in general® However

( AB)

0 a B«A«

y .g sumption

and

Ax « x A and A*y « J M

y’Ax « y

0 x A »

x? A9 y » x

s y / f * *

3: .oe these quantities art) scalars they are not altered

by talcing the transpose® Sinnc*

(x # A4 y) * l .'r

5 Ax) *

• we have

x 3 A s y s y

8 Ax

and therefor©

i<f G« f

x5 y J* » y

s x A o

A y

8 x » x 8 y * or

A yA<- & Consider next the case of hermitIan matrices when

A « A 9 . We then haY©

Theorem 2 0 A heraltian matrix has only real characteristic roots®
..v »r> V *•* *

9„ , , - .,J. P%»»

3‘ 5 * Jo S

v/'-f «w V a, •

m»ii

5na

3o*U8

.I®#

ftTOfl aw

•* .«

«• ,

3

*' riSv)

'.vX“

t“ (-* '
Proof. Let A be a characteristic root of the hsrmitian

matrix A with, x &e a corresponding vector. Since A la

also a root of A s (- A) with vector x we have by Theorem 1

A o X

s X s X8 X O Since x 8 x is a real number 4 0 we have

A 13 TF*

In [1] It was shown that the 2x2 quaternion matrix

(& ) has the root / =* a x 4- a wh®K x Is a solution

jL2&» 2JL JL ^^ of x„a_x_ » a__x_ - x a__ + a and (X s 05 is a oorres1 21 1 XI i X 22 12

ponding vector. Assume further, that a . ..

. « 0 It can

bs verified easily that A ie real* By assumption a

22

Is real 6 Next we show that ^ * a^x^ Is real* For this

purpose multiply the conjugate of x,& x by x. on the

righto Th© result ie:

Vl2 hh “ (airS22 5 Vl + a

21

X

l

•

Since (&,_ - a^.Jx x_ Is real we see that has the

11 h I i

property that

<2*4 r t r oC

when r and 2% are real numbers, r )> Oo Heno©

r 4* p ss «<.

«L and (04 o{J r « -(X,-

Since r ^

0 we have ^4 ^ ^ 0 Next we consider F, the field of values of the nxn

quaternion matrix % i o e * ? the set of quaternions

-3“
• /

p-

\

i

* m r

t

i 3

r
JL waere (x.

? , o-.c||E } Is a quaternion vector with

“ ° ^ or* thi s set of quaternions the

following theorem holds

;

Theorem 3. F Ie unaltered If the* matrix A Is replaced

by S AS irh m S * (S^) is a unitary matrix* It contains

all the characteristic roots. If^is a quaternion In F

and f an arbitrary quaternion then

J

> "1

o< ^is ax s0 in Fo

ihe field of values 1# bounded and closed.

Proof. Let s be unitary; then ST1 = 8, The element in

the- 1-th row and k-th column of S

-iA3 is then T T a S

and the field of values of the matrix S“

X

AS is therefor©

the set of numbersfi^^s^ - 1. This

can be shown to be a set of numbers in the set of values

oa th„ ut.tr xx A, namely tne values corresponding to the

vectors X s n> Since S is unitary

these vectors are unit waters fdr lx s„ S, x, =1.

. «** k Jtk 4

7k &

Since with S also s

* ie unitary it follows that

conversely the field of values of S~

X

AS Ie contained in

th© field cf values of a. This proves the first 'part of

theorem 3*

The next part is proved in the following way • Let

/I be any characteristic root of A and x a corresponding

vector. Let x ® ® ®

»

P x

r

,) ® W® may assume x a unit

vector since otherwise every component could be divided

by the real scalar and a unit vector would

thug be obtain©do Consider then the numbers In F

obtained from that particular sector xa It is2- YiA 88 A * Ac. Hence A lies In F*

The next assertion is proved by considering

C“ -

“ °

fc““- x

i

a

jj£

Xh

® " op 8Qm® vector x0 Let b© an

arbitrary quaternion and norm f « r

2

. Consider then the

vector

|J f/r, • •• J"/r Instead of x 9 . cc , s x^, It

is still a unit vector. The number in F which corresponds

to it is f

3“°i * Hence P Is also In F„'

That F Is bounded and closed follows from the fact

tet It is a continuous mapping of the set of vectors

V 6 °°" Xn

with £~ x

i

x

i " 1& hmoe ot the * ®~ dimensional ' salt sjber©.,,

[13

[23

H* ?• Le?* Eigenvalues and canonical forms of matrices with quaternion coefficients, Proc' Academy Lll, 253-260 ( 1949 )/

Royal Irish

Brenner, Matrices of Quaterliorie B Pacific Journal of Mathematics

to appear in

national Bureau of Standards

¥a ehingt on, D„ 0.

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