Matrix-valued (vector-valued) signals are everywhere these

days, such as, videos, multi-spectral images, signals from

multiarray multisensors, and high dimensional data. For these

signals, there are correlations not only over the time but also

across their matrix components. How to efﬁciently represent

and process them plays an important and fundamental role in

data science.

In [1], [2], a matrix-valued inner product was introduced for

matrix-valued (or vector-valued) signals. It leads to a weaker

orthogonality (called Orthogonality B) than the component-

wise orthogonality (called Orthogonality A) in orthogonal

multiwavelet constructions [11], [12]. The weaker orthog-

onality, i.e., Orthogonality B, provides an easier sufﬁcient

condition to construct orthonormal multiwavelets with Orthog-

onality B than the necessary and sufﬁcient condition [12] to

construct orthonormal multiwavelets with Orthogonality A. A

connection between multiwavelets and matrix-valued/vector-

valued wavelets can be found in [2]. After the works in [1],

[2], there have been many studies on matrix-valued/vector-

valued wavelets for matrix-valued/vector-valued signals in the

literature, see, for example, [3]- [10].

On the other hand, the Orthogonality B induced from the

matrix-valued inner product is stronger than the orthogonality,

which is called Orthogonality C here, induced from the

commonly used scalar-valued inner product for matrices. It is

because, as we shall see later, Orthogonality B is basically the

orthogonality between all row vectors of two matrix-valued

signals, while Orthogonality C is the orthogonality of two

long vectors of concatenated row vectors of two matrix-valued

signals. It was proved in [2] that Orthogonality B or the

matrix-valued inner product is able to completely decorrelate

matrix-valued signals not only in time domain but also across

the components inside matrix, i.e., it provides a complete

Karhunen-Lo`eve expansion for matrix-valued signals, while

Orthogonality A or Orthogonality C may not do so. In other

words, Orthogonality B induced from the matrix-valued inner

product is the proper orthogonality for matrix-valued signals.

This also means that the matrix-valued inner product is needed

to study the decorrelation of matrix-valued signals and a

conventional scalar-valued inner product may not be enough.

Since the main goal in [1], [2] was to construct orthonormal

matrix-valued (vector-valued) wavelets, not much about the

inner product or the orthogonality itself, which is more funda-

mental, was studied. In this paper, we study more propertis on

the matrix-valued inner product and its induced Orthogonality

B for matrix-valued signal space proposed in [1], [2]. We ﬁrst

deﬁne a different norm for matrix-valued signals than that

deﬁned in [2] and prove that these two norms are equivalent.

The norm deﬁned in this paper is consistent with the matrix-

valued inner product similar to that for a scalar-valued inner

product. We introduce a new linear independence concept for

matrix-valued signals and present some related properties. We

then present the Gram-Schmidt orthonormalization procedure

for a set of linearly independent matrix-valued signals. We ﬁ-

nally deﬁne matrix-valued lattices, where the newly introduced

Gram-Schmidt orthogonalization might be applied. Due to the

noncommuntativity of matrix multiplications, these concepts

and properties for matrix-valued signals and/or inner product

are not straightforward extensions of the conventional ones for

scalar-valued signals and/or inner product.

The remainder of this paper is organized as follows. In

Section II, we introduce matrix-valued signal space, matrix-

valued inner product, and deﬁne a new norm for matrix-valued

signals. We present some simple properties for the matrix-

valued inner product and prove that the new norm proposed

in this paper is equivalent to that used in [2]. In Section III, we

A Matrix-Valued Inner Product for Matrix-Valued Signals and

Matrix-Valued Lattices

XIANG-GEN XIA

Fellow, IEEE

Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA

Abstract: A matrix-valued inner product was proposed before to construct orthonormal matrix-valued wavelets for matrix-

valued signals. It introduces a weaker orthogonality for matrix-valued signals than the orthogonality of all components in

a matrix that is commonly used in orthogonal multiwavelet constructions. With the weaker orthogonality, it is easier to

construct orthonormal matrix-valued wavelets. In this paper, we re-study the matrix-valued inner product more from the

inner product viewpoint that is more fundamental and propose a new but equivalent norm for matrix-valued signals. We

show that although it is not scalar-valued, it maintains most of the scalarvalued inner product properties. We introduce a

new linear independence concept for matrix-valued signals and present some related properties. We then present the Gram-

Schmidt orthonormalization procedure for a set of linearly independent matrix-valued signals. Finally we define matrix-

valued lattices, where the newly introduced Gram-Schmidt orthogonalization might be applied.

Keywords: Matrix-valued inner product, matrix-valued signal space, nondegenerate matrix-valued signals, linearly

independent matrix-valued signals, matrix-valued lattices, matrix-valued wavelets

Received: March 25, 2021. Revised: April 19, 2022. Accepted: July 16, 2022. Published: September 9, 2022.

1. Introduction

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L2(a, b;CN×N)∆

=









f(t) = 





f11(t)f12(t)··· f1N(t)

f21(t)f22(t)··· f2N(t)

···

fN1(t)fN2(t)··· fNN (t)







:fkl ∈L2(a, b),1≤k, l ≤N









.(1)

ﬁrst introduce the concepts of degenerate and nondegenerate

matrix-valued signals and then introduce the concept of linear

independence for matrix-valued signals. The newly introduced

linear independence is different from but consistent with the

conventional one for vectors. We also present some interesting

properties on the linear independence and the orthogonal-

ity. We ﬁnally present the Gram-Schmidt orthonormalization

procedure for a set of linearly independent matrix-valued

signals, which has the similar form as the conventional one

for vectors but not a straightforward generalization due to the

noncommutativity of matrix multiplications and the matrix-

valued inner product used in the procedure. In Section IV, we

deﬁne matrix-valued lattices. In Section V, we conclude this

paper.

We ﬁrst introduce matrix-valued signal space studied in

[1], [2]. Let CN×Ndenote all N×Nmatrices of complex-

valued entries, and for −∞ ≤ a < b ≤ ∞, let L2(a, b)

denote all the ﬁnite energy signals in the interval (a, b)and

L2(a, b;CN×N)be deﬁned in (1) at the top of this page.

We call L2(a, b;CN×N)a matrix-valued signal space and

f(t)∈L2(a, b;CN×N), or simply f∈L2(a, b;CN×N), a

matrix-valued signal.

For any A∈CN×Nand f∈L2(a, b;CN×N), the products

Af,fA∈L2(a, b;CN×N). This implies that the matrix-

valued signal space L2(a, b;CN×N)is deﬁned over CN×N

and not simply over C. For f∈L2(a, b;CN×N), its integration

af(t)dt is deﬁned by the integrations of its components, i.e.,

af(t)dt =Rb

afkl(t)dt.

Let k · kMdenote a matrix norm on CN×N, for example,

the Frobenius norm,

kAkM=kAkF=



k,l=1 |Akl|2



1/2

where A= (Akl). For each f∈L2(a, b;CN×N), let kfkM

denote the norm of fassociated with the matrix norm k · kM

kfkM

∆

=



Zb

f(t)f†(t)dt



1/2

,(2)

where †denotes the complex conjugate transpose. Note that

the norm kfkof fdeﬁned in [2] has the following form

kfk∆

= Zb

akf(t)k2

Mdt!1/2

,(3)

where kf(t)kMis the matrix norm of matrix f(t)for a ﬁxed t.

We will show later that the above two norms kfkand kfkMare

equivalent in the sense that there exist two positive constants

C1>0and C2>0such that

C1kfk ≤ kfkM≤C2kfk,for any f∈L2(a, b;CN×N).(4)

We next deﬁne matrix-valued inner product for matrix-

valued signals in L2(a, b;CN×N). For two matrix-valued sig-

nals f,g∈L2(a, b;CN×N), their matrix-valued inner product

(or simply inner product) hf,giis deﬁned as the integration

of the matrix product f(t)g†(t), i.e.,

hf,gi∆

=Zb

f(t)g†(t)dt. (5)

With the deﬁnition (5), most properties of the conventional

scalar-valued inner product hold for the above matrix-valued

inner product. For instance, the following properties of the

matrix-valued inner product are clear:

(i) hf,gi=hg,fi†.

(ii) hf,fi= 0 if and only if f= 0.

(iii) kfkM=khf,fik1/2

(iv) For any A, B ∈CN×N,hAf, Bgi=Ahf,giB†.

Note that Property (iii) may not hold for the norm (3) used in

[2].

Two matrix-valued signals fand gin L2(a, b;CN×N)are

called orthogonal if hf,gi= 0. A set of matrix-valued

signals is called an orthogonal set if any two distinct matrix-

valued signals in the set are orthogonal. A sequence Φk(t)∈

L2(a, b;CN×N),k∈Z, is called an orthonormal set in

L2(a, b;CN×N)if

hΦk,Φli=δ(k−l)IN, k, l ∈Z,(6)

where δ(k) = 1 when k= 0 and δ(k) = 0 when k6= 0,

INis the N×Nidentity matrix. Due to (i) above, the

orthogonality/orthonormality between fand gis commutative,

i.e., if fand gare orthogonal/orthonormal, then gand fare

orthogonal/orthonormal too.

A sequence Φk(t)∈L2(a, b;CN×N),k∈Z, is called an

orthonormal basis for L2(a, b;CN×N)if it satisﬁes (6), and

moreover, for any f∈L2(a, b;CN×N)there exists a sequence

of N×Nconstant matrices Fk,k∈Z, such that

f(t) = X

k∈Z

FkΦk(t),for t∈[a, b],(7)

or simply

f=X

k∈Z

FkΦk,

where Fk=hf,Φki, the multiplication FkΦk(t)for each ﬁxed

tis the N×Nmatrix multiplication, and the convergence for

the inﬁnite summation is in the sense of the norm k·kMdeﬁned

2. Matrix-valued Signal Space

and Matrix-valued Inner Product

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by (2) for the matrix-valued signal space. The corresponding

Parseval equality is

hf,fi=X

k∈Z

FkF†

k.(8)

With the norm k · kMin (2), it is clear that for any el-

ement Φkin an orthonormal set in L2(a, b;CN×N), we

have kΦkkM= 1, which is consistent with the conventional

relationship between vector norm and vector inner product.

However, this property may not hold for the norm k·k in (3)

used in [2]. We refer to [2] for the Karhunen-Lo`eve expansion

with an orthonormal basis for random processes of matrix-

valued signals.

We next show the equivalence (4) of the two norms k·kM

in (2) and k · k in (3).

Proposition 1: The norms k·kMin (2) and k· k in (3) are

equivalent.

Proof: It is known that all matrix norms for constant

matrices are equivalent. Hence, we show (4) only for the

Frobenius norm, i.e., k · kM=k · kF. In this case,

kfk4

k,l=1 Zb

m=1

fkm(t)f∗

lm(t)dt

k=1 

m=1 Zb

a|fkm(t)|2dt

k6=l=1 

m=1 Zb

fkm(t)f∗

lm(t)dt

≥

k=1 

m=1 Zb

a|fkm(t)|2dt

≥1

N

Zb

k,l=1 |fkl(t)|2dt



Nkfk4.

Thus, we have

N1/4kfk ≤ kfkM.(9)

On the other hand,

kfk4

k,l=1 Zb

m=1

fkm(t)f∗

lm(t)dt

≤N

k,l=1

m=1 Zb

fkm(t)f∗

lm(t)dt

≤N

k,l=1

m=1 Zb

a|fkm(t)|2dt Zb

a|flm(t)|2dt

≤N

k,l=1

m=1 

 Zb

a|fkm(t)|2dt!2

+ Zb

a|flm(t)|2dt!2



≤N2

k=1

m=1 Zb

a|fkm(t)|2dt!2

≤N2

Zb

k,m=1 |fkm(t)|2dt



=N2kfk4.

This shows that

kfkM≤√Nkfk.(10)

Combining (9) and (10), the equivalence (4) with C1=N−1/4

and C2=N1/2between the norms k · kMin (2) and k · k in

(3) is proved. q.e.d.

Due to the equivalence of the norm k · kMproposed in

this paper and the norm k · k used in [2], all the results on

orthonormal matrix-valued wavelets obtained in [2] hold, when

the norm k·kMfor matrix-valued signals in this paper is used.

As a remark, the conventional inner product for two matrices

Aand Bis the scalar-valued inner product tr(AB†)where

tr stands for the matrix trace. It is not hard to see that with

this scalar-valued inner product, the orthogonality between two

matrix-valued signals, which is called Orthogonality C, is the

orthogonality of two long vectors of concatenated row vectors

of two matrix-valued signals. As mentioned in Introduction,

and it is also not hard to see from the above deﬁnition, the

orthogonality (6) induced from the matrix-valued inner product

in this paper for two matrix-valued signals is the orthogonality

between any row vectors including the row vectors inside

a matrix of the two matrix-valued signals, which is named

Orthogonality B in [2]. Clearly Orthogonality B is stronger

than Orthogonality C, while it is weaker than the component-

wise orthogonality called Orthogonality A in [2], commonly

used in multiwavelets [11], [12].

With Orthogonality A, a necessary and sufﬁcient condition

to construct orthonormal multiwavelets was given in [12] that

is not easy to check. However, with Orthogonality B, an easy

sufﬁcient condition to construct orthonormal multiwavelets

was obtained in [2]. Furthermore, it was shown in [2] that the

matrix-valued inner product (5) and its induced Orthogonality

B provide a complete decorrelation of matrix-valued signals

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along time and across matrix components, i.e., a complete

Karhunen-Lo`eve expansion for matrix-valued signals can be

obtained. This may not be possible for Orthogonality A or

Orthogonality C induced from a scalar-valued inner product

[13], [14]. In other words, the matrix-valued inner product (5)

is fundamental to study matrix-valued signals.

Let us ﬁrst introduce degenerate and linearly independent

matrix-valued signals, and study their properties.

A matrix-valued signal fin L2(a, b;CN×N)is called de-

generate signal if hf,fidoes not have full rank, otherwise it

is called nondegenerate signal. A sequence of matrix-valued

signals fkin L2(a, b;CN×N),k= 1,2, ..., K, are called

linearly independent if the following condition holds: if

k=1

Fkfk

∆

=f(11)

for constant matrices Fk∈CN×N,k= 1,2, ..., K, is

degenerate, then the null space of matrix F†

kincludes the null

space of matrix hf,fifor every k,k= 1,2, ..., K. Clearly,

the above linear independence returns to the conventional one

when all the above matrices are diagonal. Furthermore, if

f= 0 in (11), the above condition implies that all Fk= 0,

k= 1,2, ..., K, since in this case, the null space of hf,fiis

the whole space CN×N. This concides with the condition of

the conventional linear independence.

Proposition 2: If matrix-valued signals fk,k= 1,2, ..., K,

are linearly independent, then, all signals fk,k= 1,2, ..., K,

are nondegenerate.

Proof: Without loss of generality, assume f1is degenerate.

Let F1=INand Fk= 0 for k= 2,3, ..., K. Then, we have

that K

k=1

Fkfk=f1

is degenerate, while the null space of F†

1is 0only and does

not include the null space of hf1,f1i. In other words, fk,k=

1,2, ..., K, are not linearly independent. This contradicts the

assumption in the proposition and therefore the proposition is

proved. q.e.d.

As one can see, the above concept of degenerate signal is

similar to that of 0in the conventional linear dependence or

independence.

Proposition 3: Let Gk∈CN×N,k= 1,2, ..., K, be K

constant matrices and at least one of them have full rank.

If matrix-valued signals fk,k= 1,2, ..., K, are linearly

independent, then PK

k=1 Gkfkis nondegenerate.

Proof: Without loss of generality, let us assume G1has full

rank. If PK

k=1 Gkfk=gis degenerate, then by the linear

independence of fk,k= 1,2, ..., K, the null space of G†

cannot only contain 0, which contradicts the assumption that

G1has full rank. q.e.d.

It is clear to see that Proposition 2 is a special case of

Proposition 3.

Proposition 4: If matrix-valued signals fk,k= 1,2, ..., K,

are linearly independent, then for any full rank constant

matrices Gk∈CN×N,k= 1,2, ..., K, matrix-valued signals

∆

=Gkfk,k= 1,2, ..., K, are also linearly independent.

Proof: For any constant matrices Fk∈CN×N,k=

1,2, ..., K, if

k=1

Fkgk=

k=1

FkGkfk=f

is degenerate, then for each k,1≤k≤K, the null space of

matrix (FkGk)†=G†

kF†

kincludes the null space of matrix

hf,fi, since fk,k= 1,2, ..., K, are linearly independent.

Because all matrices Gk,k= 1,2, ..., K, have full rank, for

each k,1≤k≤K, the null spaces of F†

kand G†

kF†

kare the

same, thus, the null space of F†

kincludes the null space of

hf,fias well. This proves the proposition. q.e.d.

Similar to the conventional linear dependence of vectors,

we have the following result for matrix-valued signals.

Proposition 5: For a matrix-valued signal f∈

L2(a, b;CN×N)and two constant matrices A, B ∈CN×N,

matrix-valued signals Afand Bfare linearly dependent.

Proof: If Afand Bfare linearly independent, then, from

Proposition 2 it is easy to see that matrices Aand Ball have

full rank and fis nondegenerate. Then, we have

BA−1Af−Bf= 0,

which contradicts with the assumption of the linear indepen-

dence of Afand Bf. This proves the proposition. q.e.d.

Although it is obvious for the conventional vectors, the

result in Proposition 5 for matrix-valued signals may not be

so, due to the matrix-valued coefﬁcient multiplications as it

can be seen from the above proof. We next consider more

general linear combinations of linearly independent matrix-

valued signals.

For 1≤p≤K, let S1, ..., Spbe a partition of the index

set {1,2, ..., K}and each Sihas Kielements, where Si1∩

Si2=∅for 1≤i16=i2≤p,∪p

i=1Si={1,2, ..., K}, and

1≤K1, ..., Kp≤Kwith K1+K2+···+Kp=K.

Proposition 6: For each i,1≤i≤p, let Gki∈CN×N,

ki∈Si, be Kiconstant matrices and at least one of them

have full rank. If matrix-valued signals fk,k= 1,2, ..., K,

are linearly independent, then the following pmatrix-valued

signals: X

ki∈Si

Gkifki,for i= 1,2, ..., p,

are linearly independent.

Proof: Let Fi∈CN×N,i= 1,2, ..., p, be constant matrices.

Assume that p

i=1

FiX

ki∈Si

Gkifki=g

is degenerate. Then,

i=1 X

ki∈Si

FiGkifki=g,

3. Linear Independence and

Gram-schmidt Orthonormalization

3.1 Degenerate and Linearly

Independent Matrix-valued Signals

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and by the linear independence of fk,k= 1,2, ..., K, we know

that the null space of (FiGki)†for every ki∈Siand every

i= 1, ..., p contains the null space of matrix hg,gi. From the

condition in the proposition, without loss of generality, we

may assume that Gki,1, for some ki,1∈Si, has full rank for

1≤i≤p. Thus, the null space of (FiGki,1)†, or G†

ki,1F†

contains the null space of hg,gifor 1≤i≤p. Since Gki,1

has full rank, the null space of F†

imust contain the null space

of hg,gifor 1≤i≤p. This proves the proposition. q.e.d.

Note that when p= 1 in Proposition 6, it returns to

Proposition 3, and when p=Kin Proposition 6, it returns to

Proposition 4.

Proposition 7: If fk,k= 1,2, ..., K, form an orthonormal

set in L2(a, b;CN×N), then, they must be linearly indepen-

dent.

Proof: For constant matrices Fk∈CN×N,k= 1,2, ..., K,

let K

k=1

Fkfk=f.

Then, from the Parseval equality (8), we have

k=1

FkF†

k=hf,fi.

Assume that for some vector u6= 0, we have hf,fiu= 0 but

F†

k0u6= 0 for some k0,1≤k0≤K. Then,

0< u†Fk0F†

k0u≤

k=1

u†FkF†

ku=u†hf,fiu= 0,

which leads to a contradiction. Thus, for each k,1≤k≤K,

the null space of F†

kincludes the null space of hf,fi. This

proves the linear independence of fk,k= 1,2, ..., K.q.e.d.

Corollary 1: Assume fk,k= 1,2, ..., K, are nondegen-

erate matrix-valued signals and form an orthogonal set in

L2(a, b;CN×N). Then, gk

∆

=hfk,fki−1/2fk,k= 1,2, ..., K,

form an orthonormal set in L2(a, b;CN×N), and fk,k=

1,2, ..., K, are linearly independent.

Proof: Since all matrix-valued signals fkare nondegenerate,

matrices hfk,fkiall have full rank. From the property (iv) for

the matrix-valued inner product and {fk}is an orthogonal set,

for every k, l,1≤k, l ≤K,

hgk,gli=hfk,fki−1/2hfk,flihfl,fli−1/2=δ(k−l)IN.

Thus, gk,k= 1,2, ..., K, form an orthonormal set in

L2(a, b;CN×N).

Then, the linear independence of fk=hfk,fki1/2gk,k=

1,2, ..., K, immediately follows from Propositions 4 and 7.

q.e.d.

The result in Corollary 1 is consistent with the conventional

one for vectors, i.e., any orthogonal set of nonzero vectors

must be linearly independent. However, there is a difference.

In the above relationship between orthogonality and linear

independence, matrix-valued signals need to be nondegenerate.

Note that it is possible that a matrix-valued signal fin an

orthogonal set in L2(a, b;CN×N)is degenerate, i.e., hf,fi

may not necessarily have full rank, even though f6= 0. Thus,

a general orthogonal set of matrix-valued signals may not

have to be linearly independent. This does not occur for any

orthogonal set of nonzero signals when a scalar-valued inner

product is used.

We are now ready to present the Gram-Schmidt orthonor-

malization for a ﬁnite sequence of linearly independent matrix-

valued signals. Let fk∈L2(a, b;CN×N),k= 1,2, ..., K, be

linearly independent. The Gram-Schmidt orthonormalization

for this sequence is as follows, which is similar to, but not a

straightforward extension of, the conventional one, due to the

noncommutativity of matrix multiplications.

Since fk∈L2(a, b;CN×N),k= 1,2, ..., K, are linearly

independent, by Proposition 2, f1is nondegenerate, i.e., matrix

hf1,f1iis invertible and positive deﬁnite. Let

g1=hf1,f1i−1/2f1.(12)

Then, we have

hg1,g1i=hf1,f1i−1/2Zb

f1(t)f†

1(t)dthf1,f1i−1/2

=hf1,f1i−1/2hf1,f1ihf1,f1i−1/2=IN.(13)

Let

g2=f2− hf2,g1ig1,(14)

g2=hˆ

g2,ˆ

g2i−1/2ˆ

g2.(15)

For (15) to be vaild, we need to show that ˆ

g2in (14) is

nondegenerate. In fact, if ˆ

g2is degenerate, then, from (14)

and (12), we have

g2=f2− hf2,g1ihf1,f1i−1/2f1

=F1f1+F2f2,

where F2=INand F1=−hf2,g1ihf1,f1i−1/2. Similar to the

proof of Proposition 2, this contradicts the assumption that f1

and f2are linearly independent. Therefore, it proves that ˆ

in (14) is nondegenerate and (15) is well-deﬁned.

Let us then check the orthogonality between g1and ˆ

g2.

From (14) and (13), we have

hˆ

g2,g1i=hf2,g1i − hf2,g1ihg1,g1i

=hf2,g1i − hf2,g1i= 0.

From (15) and (13), we have that g1and g2form an orthonor-

mal set.

Repeat the above process and for a general k,2≤k≤K,

we let

gk=fk−

k−1

l=1 hfk,gligl,(16)

gk=hˆ

gk,ˆ

gki−1/2ˆ

gk.(17)

With the same proof as the above g1and g2, we have the

following proposition.

Proposition 8: For a linearly independent set of matrix-

valued signals fk,k= 1,2, ..., K, let g1,g2, ..., gKbe con-

structed in (12) and (14)-(17). Then, g1,g2, ..., gKform an

orthonormal set.

3.2 Gram-schmidt Orthonormalization

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As we can see, although the above Gram-Schmidt orthonor-

malization procedure for matrix-valued signals is similar to

the conventional one for vectors, it is not a straightforward

generalization due to 1) the noncommuntativity of matrix

mulitpications and 2) the matrix-valued inner product used

in the above procedure.

We also want to make a comment on the nondegenerate and

linear independence for matrix-valued signals. The condition

for nondegenerate matrix-valued signals is a weak condition.

Unless the row vectors of functions are linearly dependent

in the conventional sense, otherwise, a matrix-valued signal

is usually nondegenerate. For a ﬁnite set of nondegenerate

matrix-valued signals, they usually satisfy the condition for

linear independence for matrix-valued signals deﬁned above,

i.e., they are usually linearly independent and therefore, they

can be made to an orthonormal set by using the above Gram-

Schmidt procedure.

Another comment on the linear independence for matrix-

valued signals is that the deﬁnition in (11) is only for left mul-

tiplication of constant matrices Fkto matrix-valued signals fk.

Similar deﬁnitions for linear independences of matrix-valued

signals with right constant matrix multiplications and/or mixed

left and right constant matrix multiplications may be possible.

Although what is studied in this paper is for continuous-time

matrix-valued signals, it can be easily generalized to discrete-

time matrix-valued signals (sequences of ﬁnite or inﬁnite

length).

In this section, based on the matrix-valued signal space with

the matrix-valued inner product, we introduce matrix-valued

lattices.

We ﬁrst introduce matrix-valued lattices. For convenience,

in what follows we only consider the Frobenius norm for

matrices, i.e., k · kM=k · kF, and real matrix-valued signal

spaces, and also let Rdenote the real matrix-valued signal

space R∆

=L2(a, b;RN×N). Let ZN×Ndenote all N×N

matrices of integer entries.

For a ﬁnite many linearly independent real matrix-valued

signals fk∈ R,k= 1,2, ..., K, let RKdenote the matrix-

valued signal space linearly expanded by them, i.e.,

RK=(K

k=1

Fkfk:Fk∈RN×N, k = 1,2, ..., K).(18)

From what was studied in the previous section, clearly, fk,

k= 1,2, ..., K, form a basis in RK. The matrix-valued lattice

formed by this basis in RKis deﬁned as

L=(K

k=1

Fkfk:Fk∈ZN×N, k = 1,2, ..., K),(19)

which is a subset/subgroup of RK. The basis fk,k=

1,2, ..., K, is called a basis for the Kdimensional matrix-

valued lattice L.

The fundamental region of this lattice Lcan be deﬁned

similar to the conventional lattice as follows. A set F ⊂ RK

is called a fundamental region, if its translations x+F=

{x+f:f∈ F} for x∈ L form a partition of RK. Since the

basis elements fk,k= 1,2, ..., K, are not constant real vectors

as in the conventional lattices, it would not be convenient to

deﬁne the determinant of the lattice. However, with the Gram-

Schmidt orthonormalization developed in the previous section,

we may deﬁne the determinant of the lattice directly as

det(L) =

k=1 kˆ

fkkF,(20)

where ˆ

fk,k= 1,2, ..., K, are from the following Gram-

Schmidt orthogonalization of fk,k= 1,2, ..., K, which is

from the Gram-Schmidt orthonormalization in the previous

section: ˆ

f1=f1,

fk=ˆ

gk=fk−

k−1

l=1

µl,kˆ

fl,(21)

where

µl,k =hfk,ˆ

flihˆ

fl,ˆ

fli−1, l = 1,2, ..., k−1and k= 2,3, ..., K.

(22)

It is clear to see that the spaces linearly spanned by

{f1,f2, ..., fK}and {ˆ

f1,ˆ

f2, ..., ˆ

fK}are the same, i.e., RKin

(18), since they can be linearly (over RN×N) represented by

each other similar to the conventional vectors.

From the Gram-Schmidt orthogonalization (21), we have

hfk,fki=hˆ

fk,ˆ

fki+

k−1

l=1

µl,khˆ

fl,ˆ

fliµ†

l,k,(23)

for k= 1,2, ..., K. Using Property (iii) in Section II, the

identity (23) implies

kfkk2

F≤ kˆ

fkk2

F+

k−1

l=1 kµl,kk2

F· kˆ

flk2

F,(24)

for k= 1,2, ..., K. From (23), it is also clear that kfkkF≥

kˆ

fkkF,k= 2, ..., K.

As a remark, we know that the conventional Gram-Schmidt

orthogonalization plays an important role in the LLL algorithm

for the conventional lattice basis reduction [20]. It is, however,

not clear how the Gram-Schmidt orthogonalization for matrix-

valued signals introduced above can be applied in matrix-

valued lattice basis reduction.

In this paper, we re-studied the matrix-valued inner product

previously proposed to construct orthonormal matrix-valued

wavelets for matrix-valued signal analysis [1], [2] where

not much on the matrix-valued inner product or its induced

Orthogonality B, which is more fundamental, was studied.

In order to study more on the matrix-valued inner product

and its induced Orthogonality B, we ﬁrst proposed a new

norm for matrix-valued signals, which is more consistent with

the matrix-valued inner product than that used in [2], and

is similar to that with the conventional scalar-valued inner

product. We showed that these two norms are equivalent,

which means that with the newly proposed norm, all the

4. Matrix-valued Lattices

5. Conclusion

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results for contructing orthonormal matrix-valued wavelets

obtained in [2] still hold. We then proposed the concepts

of degenerate and nondegenerate matrix-valued signals and

deﬁned the linear independence for matrix-valued signals,

which is different from but similar to the conventional linear

independence for vectors. We also presented some properties

on the linear independence and the orthogonality. We then

presented the Gram-Schmidt orthonormalization procedure for

a set of linearly independent matrix-valued signals. Although

this procedure is similar to the conventional one for vectors,

due to the noncommutativity of matrix multiplications and the

matrix-valued inner product used in the procedure, it is not

a straightforward generalization. We ﬁnally deﬁned matrix-

valued lattices, where the newly introduced Gram-Schmidt

orthogonalization might be applied.

Since it was shown in [2] that the matrix-valued inner

product and Orthogonality B provide a complete Karhunen-

Lo`eve expansion for matrix-valued signals, which a scalar-

valued inner product may not do, it is believed that what

was studied in this paper for matrix-valued inner product for

matrix-valued signal space will have fundamental applications

for high dimensional signal analysis in data science.

As a ﬁnal note, after this paper was written, it has been

found that the matrix-valued signal space with the matrix-

valued inner product in this paper is related to Hilbert modules,

see, for example, [15]- [19]. Interestingly, it was mentioned

in [18] that there does not exist any general notion of “C∗-

linear independence” due to the existence of zero-divisors. We

believe that the linear independence for matrix-valued signals

introduced in this paper is novel.

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References

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