# Note on traces of matrix products involving inverses of positive definite ones

### Andrzej Z. Grzybowski

Journal of Applied Mathematics and Computational Mechanics |
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NOTE ON TRACES OF MATRIX PRODUCTS INVOLVING INVERSES OF POSITIVE DEFINITE ONES

Andrzej Z. Grzybowski

Institute of Mathematics, Czestochowa
University of Technology

Czestochowa, Poland

andrzej.grzybowski@im.pcz.pl

Received: 1 December 2017;
Accepted: 7 February 2018

**Abstract.**
This short note is devoted to the analysis of the trace of a product of two
matrices in the case where one of them is the inverse of a given positive
definite matrix while
the other is nonnegative definite. In particular, a relation between the trace
of **A**^{–1}**H** and
the values of diagonal elements of the original matrix **A** is analysed.

*MSC 2010:** 15A09, 15A42, 15A63 *

*Keywords:** matrix product, trace inequalities,
inverse matrix *

1. Introduction

Traces of matrix products are of special
interest and have a wide range of appli-
cations in different fields of science such
as economics, engineering, finance, hydro-
logy and physics. They also arise naturally in the applications of
mathematical sta-
tistics, especially in regression analysis, [1-3] or the analysis of
discrete-time statio-
nary processes [4]. There are papers devoted to the role of
matrix-product-traces
in the description of the probability distributions of quadratic forms of
random vectors, [1, 5], or to the development of approximate boundaries
for their (i.e. product traces) values
[6-11]. In some statistical applications the product under consideration
involves inverse **A**^{–1} of a given positive definite matrix **A**.
In particular, it takes place in the Bayesian analysis in regression modelling,
where the matrix **A** can be interpreted as the covariance matrix of the
disturbances and/or a priori distribution of unknown system-parameters [2, 3].

In this paper, we present an equation
concerning traces of certain matrix products involving an inverse **A**^{–1
}of a given matrix **A**, and next this equation is used to obtain a
result relating the changes in the values of the diagonal elements of the
original matrix **A** with the values of the considered trace. This note is
organized
as follows. In the next section we recall some definitions and facts that will
be
necessary to state and prove the new results. In Section 3, we state the main
equation and then, in Section 4 we present some possible applications.

2. Preliminary definitions, facts and notation

All of the matrices considered here are
real. For any square matrix **A** = [*a _{ij}*]

_{n}_{´n}the symbol

_{ }denotes the cofactor of the element

*a*, and adj(

_{ij}**A**) denotes the transpose of a matrix with elements being the cofactors of appropriate elements of

**A**, i.e. adj(

**A**) = []

^{T}.

For any square matrix **A** we write **A** > 0
(or **A** ³ 0) if the matrix is positive
definite (or positive
semi-definite), i.e. **A** is symmetric and **x ^{T}Ax** > 0 for
all nonzero column vectors

**x**Î

**R**

*(or*

^{n}**x**³ 0 for all

^{T}Ax**x**Î

**R**

*). A square matrix is nonnegative definite if it is positive definite or a positive semi-definite one.*

^{n}The following facts concerning determinants and/or inverses of matrices expressed in so-called block forms can be found in various textbooks, see e.g. [12, pp. 33-34].

**Fact 1.**

Let **A** = [*a _{ij}*]

_{k}_{´k}and

**x**be a

*k*-dimensional column vector. Then for any number

*a*

| (1) |

**Fact 2.**

Let **A** and **B** be symmetric
matrices. The following equality is true, provided that the inverses that occur
in this expression do **exist**

(2) |

with matrices **D** and **E** being
defined as follows: **D** = **A**^{–1}**C**^{T}
and **E** = **B** – **CA**^{–1}**C**^{T}.

**Fact 3.**

Let **A** be nonsingular and let **u**,
**v** be two column vectors with dimensions equal to the order of **A**.
Then

(3) |

This useful equation gives a method of
computing the inverse of the left-hand side of (3) knowing only the inverse of **A**.

**Definition** (*Hadamard
Product*). If **A** = [*a _{ij}*]

_{n}_{´m}and

**B**= [

*b*]

_{ij}

_{n}_{´m}are the matrices of the same dimensions

*n*x

*m*, then their Hadamard product is the

*n*x

*m*- matrix

**А***

**В**of elementwise products, i.e.

**А***

**В**= [

*a*·

_{ij}*b*]

_{ij}

_{n}_{´m}

We have the following results involving Hadamard products.

**Fact 4.**

For any square matrices **A**, **B** of
the same order, the following equality holds:

Tr AB = e^{T } | (A*B^{T}) e (4) |

with **e** being the vector of an appropriate
dimension with all coefficients equal
to unity, i.e. **e**^{T }= (1,1,..,1).

**Schur’s lemma.**
Let **A** and **B** be square matrices of the same order. If these
matrices are both nonnegative definite then their Hadamard product **А*****В**
is also nonnegative definite.

3. The main result

Let us consider a square symmetric matrix of
order *k* given in the following block-form

A = | (5) |

Let matrix **C** have the following
block-form:

(6) |

where the constant *d* equals

Proposition 1

Let **A** > 0 and **H** ³ 0.
Let **A**_{11} and **H**_{11} be the submatrices obtained
by deleting the first row and the first column in **A** and **H**,
respectively. Then the matrix **C** given by (6) is positive definite, the
constant *d* in (6) is a positive one, and

Tr(A^{–1}H) – Tr(H_{11}) = Tr CH | (7) |

**Proof.**

Let us note that *a*_{11} > 0
and det(**A**_{11}) > 0 (it is because **A** > 0).
Thus the inverse of **A** can be computed with the help of the Facts 1, 2.
Indeed, let us express the matrices **D**, **E** that appears in formula
(2) using the objects from the matrix given
in (5):

**D** = **w**^{T}/*a*_{11}
and

E = – ww^{T}/*a*_{11 }

Now, by (2), the inverse of the block matrix takes on the form

(8) |

It results from the Fact 3 that we have the following equality for lower-right block in (8):

(9) |

Taking the formula (9) into account we can obtain new forms for the remaining three blocks in (8). The first one, in the upper left corner (as a matter of fact 1x1 matrix), takes on the following form:

The second block in the first row can be transformed in the following way:

Similarly the third one takes on the form:

Now the matrix **A**^{–1} can be
expressed by the following simple formula:

= + | (10) |

Thus, for any matrix **H** of an
appropriate dimension the following equalities hold:

Tr(A^{–1}H)
= Tr(CH + DH) = Tr(CH) + Tr(DH) = Tr(CH) + Tr(H_{11}) |

Now we show that the number *d* is
positive.

Indeed, in view of Fact 1 we have:

But , so we finally have:

Since the matrix **A** is positive
definite one, both above determinants are positive and this yields the
positivity of *d*:

To complete the proof we need to show that
the matrix **C** is positive definite one. For *an
arbitrary* *k*-dimensional vector **x**^{T} = (*x*_{1},**x**_{2}^{T}),
*x*_{1}ÎR, **x**_{2}ÎR^{k}^{–1}
we have

x^{T }C x = d_{ }(x_{1} – b)^{2},
with b = x_{2}. |

This yields that for any nonzero vector **x**ÎR* ^{k}*,

**x**

^{T }

**C x**> 0 and thus the matrix

**C**is positive definite.

The proof of Proposition 1 is completed.

4. Some applications

First we state a proposition which is a quite straightforward conclusion from Proposition 1.

Proposition 2

Let the matrices **A**, **H**
satisfy the assumptions from Proposition 1, and let,
as previously indicated, symbols **A**_{11} and **H**_{11}
denote the submatrices obtained by deleting the first row and the first column
in **A** and **H**, respectively. Then

tr(A^{–1}H) – tr(H_{11}) ³ 0 |

**Proof. **

From Proposition 1 we know that Tr(**A**^{–1}**H**) – Tr(**H**_{11}) = Tr **CH** and that **C**
is a positive definite matrix. From Fact 4 we have:

Tr(CH) = e^{T }(C*H^{T}) e |

By the assumptions matrix **H** is
nonnegative definite, thus **H**^{T} is also nonnegative definite.
From nonnegative definiteness of the matrices **C** and **H**^{T}
it follows,
in the light of the Schur’s lemma, that **C*****H**^{T} is nonnegative definite as well, and
consequently **e**^{T }(**C*****H**^{T})** e** ³ 0. This
fact completes the proof.

Let us consider now a symmetric positive
definite matrix **A** = [*a _{ij}*]. Let us define a
matrix

**A**

*= [*

_{x}*a*] related with

_{ij}**A**by the formula:

*a*

_{11 }=

*a*

_{11 }–

*f*(

*x*) and

*a*=

_{ij}*a*for all remaining elements of

_{ij}**A**, where,

*D*Í

**R**, is a given real function.

Proposition 3

Let **A** > 0 and **H** ³ 0 be
square matrices of the same order. Let *f* be a function differentiable on
an interval *D* such that **A*** _{x}* > 0 for
all

*x*Î

*D*. Let for all

*x*Î

*D*, a function be defined as

*T*(

*x*) = Tr(). Then

*T*is nondecreasing (non- increasing) if and only if

*f*is nondecreasing (nonincreasing).

**Proof.**

It follows from Proposition 1 that

T(x) = Tr() = Tr(H_{11}) + D(x)Tr CH/d |

where

While the matrix **C** and constant *d*
are defined in (6).

Note that the function *T* depends on
its argument only thru the function *D*.
Now a little calculation shows that for each *x*Î*Int*(*D*) the derivative of *T* does exist and can be
expressed in the following form:

In view of the assumptions about the matrices
**A** and **H** and our previous results, the ratio is nonnegative, which completes the
proof.

5. Conclusions

Due the fact that in our results the matrix **H**
is nonnegative definite, one may consider matrices of the form **H** = **ww**^{T},
with **w** being a column vector. Because of the well-known relation **wAw**^{T} = Tr(**Aww**^{T}),
it is easy to see that the above results can be also used for the analysis of
the quadratic forms **wAw**^{T} with **A** being a given
symmetric positive definite matrix.

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