New Formulation and Computation for Generalized Singular Values of Grassman Matrix Pair

1 Introduction

The generalized singular value decomposition (GSVD) used in mathematics and numerical computations

[10, 11, 22] is a very useful and versatile tool. The GSVD of two matrices having the same number of columns was first proposed by Van Loan [22]. It is very useful in many matrix computation problems and practical applications, such as the Kronecker canonical form of a general matrix pencil, the linearly constrained least-squares problem, the general Gauss-Markov linear model, the generalized total least-squares problem, real-time signal processing, comparative analysis of DNA microarrays and so on; see for example [2, 3, 4, 5, 6, 7, 8, 10, 11, 14, 18, 20, 22, 23, 24, 26].

Numerical methods and perturbation analysis of GSVD have been well developed; see for instance [4, 7, 8, 14, 18, 20, 21, 22, 23, 24, 26]. The GSVD of two matrices having the same number of columns was first proposed by Van Loan [22]. Van Loan [22] and Paige [18] provided algorithms for computing the generalized singular value decomposition. Bai and Demmel [4] described a variation of Paige’s algorithm for computing the GSVD introduced by Van Loan [22] and Paige and Saunders [19]. Ewerbring and Luk [8] and Zha [26] proposed a GSVD for matrix triplets. Stewart [20] and Van Loan [23] proposed two algorithms for computing the GSVD. On perturbation analysis of the GSVD, Sun [21] and Li [14] presented several perturbation bounds of generalized singular values (GSVs) of a Grassman matrix pair (GMP) and their associated subspaces. Xu et al. [24] provided the explicit expression and sharper bounds of the chordal metric between GSVs of a GMP.

Recently, the GSVD plays an important role in the analysis of DNA microarrays and gene data. For arbitrary generalized singular value $(α_{i}, β_{i})$ of a GMP, the ratio $α_{i} / β_{i}$ can be used in comparison analysis of gene data as an indicator; see for example [2, 3]

. Moreover, the formulation for arbitrary GSV of a GMP has not been studied. Usually, the GSVD are computed by making decompositions (e.g., QR decomposition and CS decomposition). In these cases, computing the whole decomposition makes computational cost higher. The main goal of this paper is to propose new model formulations for computing arbitrary GSV of a GMP. We first derive new formulations for computing arbitrary GSV of the GMP. By using truncated filter matrices, the matrix optimization problems can be reformulated to locate the

i

-th GSV of the GMP. The resulting optimization problems can be solved by using Newton’s method on Grassmann manifold. Numerical examples on synthetic data sets and gene expression data sets are reported to demonstrate the high accuracy and the fast computation of the proposed method for computing arbitrary GSV of a GMP.

1.1 Organization

The rest of this paper is organized as follows. In Section 2, we provide mathematical preliminaries and derive new formulations for computing GSVs of a GMP. In Section 3, we present the numerical method for solving the resulting optimization models. In Section 4, we provide numerical examples to show the efficiency of the theoretical results. Finally, some concluding remarks are given in Section 5.

1.2 Notation

Throughout this paper we always use the following notation and definitions. $ı$ denotes imaginary unit $\sqrt{- 1}$ and $R$ , $C$ , $R^{n}$ , $C^{m \times n}$ and $U_{n}$ are the sets of real numbers, complex numbers, $n$

-dimensional real vectors,

m \times n

complex matrices and

n \times n

unitary matrices accordingly.

| \cdot |

stands for the absolute value of a complex number.

I_{n}

and

O_{m \times n}

stand for the identity matrix of order

n

and the

m \times n

zero matrix, respectively.

¯ A

A^{T}

A^{H}

A^{- 1},

det (A), t r (A)

denote the conjugate, transpose, conjugate transpose, inverse, determinant and trace of a matrix

A

accordingly. By

∥ \cdot ∥_{2}

we denote the spectral norm of a matrix. The singular value set of

A

is denoted by

σ (A)

. For given matrices

A, B \in C^{n \times n}

A < (\leq) B

means

B - A

is a positive (semi-)definite matrix. For a matrix

A \in C^{n \times n}

, we denote by

σ_{1} (A) \geq σ_{2} (A) \geq \dots σ_{n} (A) \geq 0

its singular values, arranged in decreasing order. We denote

[γ]

the largest integer less than or equal to a real number

γ

. The symbol “

\otimes

” means the Kronecher product and

v e c (\cdot)

creates a column vector from a matrix by stacking its column vectors below one another. Finally,

d i a g (A, B)

denotes a block diagonal matrix in which the diagonal blocks are square matrices

A

and

B

Definition 1.1

[10] Let $A \in C^{m \times n}$ and $B \in C^{p \times n}$ . A matrix pair ${A, B}$ is an $(m, p, n)$ -GMP if rank $(A^{T}, B^{T})^{T} = n$ .

For any $(m, p, n)$ -GMP ${A, B}$ , one may see $Z = (A^{T}, B^{T})^{T}$ as a point of the complex projective space $G_{n, m + p}$ of all $n$ -dimensional subspaces of the $(m + p)$ -dimensional complex space $C^{m + p}$ , i.e., one can identify $Z$ with the linear subspace $R (Z)$ (see e.g. [15, 17, 21]). Clearly, if ${A, B}$ is an $(m, p, n)$ -GMP, then $(A^{H} A, B^{H} B)$ is a definite matrix pair, i.e., $x^{H} A^{H} A x + x^{H} B^{H} B x > 0$ for all nonzero $x \in C^{n}$ . The definite pair $(A^{H} A, B^{H} B)$ has $n$

generalized eigenvalues, and thus the GMP

{A, B}

has

n

generalized singular values. A well-developed perturbation theory for the generalized eigenvalue problem of definite pencils is available. Perturbation bounds for the generalized singular value problem can be obtained with the help of the close relation between the two problems. However, the bounds obtained in this way are often unsatisfactory, just like the perturbation bounds for the singular values (SVs) of a single matrix A obtained through the perturbation bounds for the eigenvalues of the Hermitian matrix

A^{H} A

. Therefore, special attention deserves to be paid to perturbations for the generalized singular value problem. Therefore, research on numerical methods and perturbation analysis of the GSVD of a GMP is an important topic (see e.g. [4, 14, 21, 22, 24]).

Definition 1.2

[10] Let ${A, B}$ be an $(m, p, n)$ -GMP. A nonnegative number pair $(α, β)$ is a GSV of the GMP ${A, B}$ if

(α, β) \neq (0, 0), det (β^{2} A^{H} A - α^{2} B^{H} B) = 0, α, β \geq 0.

The set of GSV of ${A, B}$ is denoted by $σ {A, B} .$ Evidently,

σ {A, B} = {(α, β) \neq (0, 0) | d e t (β^{2} A^{H} A - α^{2} B^{H} B) = 0, α, β \geq 0} .

In the literature [22, 23], there are several formulations of the GSVD. Here we adopt the following form.

Definition 1.3

Let {A,B} be an (m,p,n)-GMP. Then there exist unitary matrices $U \in C^{m \times m}, V \in C^{p \times p},$ and a nonsingular matrix $R \in C^{n \times n}$ such that

U^{H} A R^{- 1} = Σ_{A}, V^{H} B R^{- 1} = Σ_{B},

(1.1)

Σ_{A} = (\begin{matrix} Λ O_{(m - r - s) \times (n - r - s)} \end{matrix}), Σ_{B} = (\begin{matrix} O_{(p + r - n) \times r} Ω \end{matrix}),

(1.2)

where

Λ = d i a g (α_{1}, \dots, α_{r + s}), Ω = d i a g (β_{r + 1}, \dots, β_{n}),

with

1 = α_{1} = \dots = α_{r} > α_{r + 1} \geq \dots \geq α_{r + s} > α_{r + s + 1} = \dots = α_{n} = 0,

0 = β_{1} = \dots = β_{r} < β_{r + 1} \leq \dots \leq β_{r + s} < β_{r + s + 1} = \dots = β_{n} = 1,

and

α_{i}^{2} + β_{i}^{2} = 1, 1 \leq i \leq n .

2 Trace function optimization model formulation

In this section we give new model formulation of any GSV of a GMP by trace function under one variable.

Lemma 2.1

[25] Let $c \in R$ and $A_{1}, \dots, A_{m} \in C^{n \times n}$ with $σ_{1} (A_{i}) \geq σ_{2} (A_{i}) \geq \dots \geq σ_{n} (A_{i})$ for $i = 1, \dots, m$ . Then we have

	$max U_{1}, \dots, U_{m} \in U_{n} R e [t r (c I_{n} \pm m \prod j = 1 U_{j} A_{j})] = n c + n \sum i = 1 m \prod j = 1 σ_{i} (A_{j}),$
	$max U_{1}, \dots, U_{m} \in U_{n} \| t r (c I_{n} \pm m \prod j = 1 U_{j} A_{j}) \| = n ∣ c ∣ + n \sum i = 1 m \prod j = 1 σ_{i} (A_{j}),$
	$min U_{1}, \dots, U_{m} \in U_{n} R e [t r (c I_{n} \pm m \prod j = 1 U_{j} A_{j})] = n c - n \sum i = 1 m \prod j = 1 σ_{i} (A_{j}),$

min U_{1}, \dots, U_{m} \in U_{n} | t r (c I_{n} \pm m \prod j = 1 U_{j} A_{j}) | = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} n ∣ c ∣ - n \sum i = 1 m \prod j = 1 σ_{i} (A_{j}), & \frac{1}{n} n \sum i = 1 m \prod j = 1 σ_{i} (A_{j}) \leq∣ c ∣, 0, & \frac{1}{n} n \sum i = 1 m \prod j = 1 σ_{i} (A_{j}) \geq∣ c ∣ . \end{matrix}

Lemma 2.2

[12] Let $f (X) : C^{n \times n} \to R$ be an analytical function of several complex variables on the domain $X X^{H} \leq I_{n}$ . Then $f (X)$ attains its maximum modulus on the characteristic manifold ${X \in C^{n \times n} : X X^{H} = I_{n}}$ .

We now give new formula model of any GSV of a GMP by trace function optimization under one variable.

Lemma 2.3

Let ${A, B}$ be an (m,p,n)-GMP and the GSV $(α_{i}, β_{i})$ of ${A, B}$ be given in (1.1) and (1.2). The following conclusions hold true.

If $n \leq m,$ then for $1 \leq i \leq n$ ,

$α_{i}^{2}$ $=$ $max Φ_{1} \in U_{m} t r (A^{H} Φ_{1}^{H} Q_{i} Φ_{1} A (A^{H} A + B^{H} B)^{- 1})$ (2.1)

$- max Φ_{1} \in U_{m} t r (A^{H} Φ_{1}^{H} Q_{i - 1} Φ_{1} A (A^{H} A + B^{H} B)^{- 1}) \equiv ϕ_{i},$

where

$Q_{i} = d i a g (T_{i}, O_{(m - n) \times (m - n)}) with T_{i} = d i a g (I_{i}, O_{(n - i) \times (n - i)}) .$ (2.2)
If $n \leq p$ , then for $1 \leq i \leq n$ ,

$β_{i}^{2}$ $=$ $max Φ_{2} \in U_{p} t r (B^{H} Φ_{2}^{H} P_{i} Φ_{2} B (A^{H} A + B^{H} B)^{- 1})$ (2.3)

$- max Φ_{2} \in U_{p} t r (B^{H} Φ_{2}^{H} P_{i - 1} Φ_{1} B (A^{H} A + B^{H} B)^{- 1}) \equiv ψ_{i},$

where

$P_{i} = d i a g (O_{(p - n) \times (p - n)}, F_{i}) with F_{i} = d i a g (O_{(i - 1) \times (i - 1)}, I_{n - i + 1}) .$ (2.4)
If $m \leq n$ , then $α_{m + 1} = \dots = α_{n} = 0$ and for $1 \leq i \leq m$ ,

$α_{i}^{2}$ $=$ $max Ψ_{1} \in U_{m} t r (A^{H} Ψ_{1}^{H} Q_{i} Ψ_{1} A (A^{H} A + B^{H} B)^{- 1})$ (2.5)

$- max Ψ_{1} \in U_{m} t r (A^{H} Ψ_{1}^{H} Q_{i - 1} Ψ_{1} A (A^{H} A + B^{H} B)^{- 1}) \equiv φ_{i},$

where

$Q_{i} = d i a g (I_{i}, O_{(m - i) \times (m - i)}) .$ (2.6)
If $p \leq n$ , then $β_{1} = \dots = β_{n - p} = 0$ and for $n - p + 1 \leq i \leq n$ ,

$β_{i}^{2}$ $=$ $max Ψ_{2} \in U_{p} t r (B^{H} Ψ_{2}^{H} P_{i} Ψ_{2} B (A^{H} A + B^{H} B)^{- 1})$ (2.7)

$- max Ψ_{2} \in U_{p} t r (B^{H} Ψ_{2}^{H} P_{i - 1} Ψ_{2} B (A^{H} A + B^{H} B)^{- 1}) \equiv χ_{i} .$

where

$P_{i} = d i a g (O_{(p - n + i - 1) \times (p - n + i - 1)}, I_{n - i + 1}) .$ (2.8)

Proof: a) If $n \leq m,$ then for $1 \leq i \leq n$ , we let $Q_{i}$ be defined by (2.2). We set $Σ_{A} = (\begin{matrix} {^Σ}_{A} O_{(m - n) \times n} \end{matrix})$ and for any $Φ_{1} \in U_{m}$ , we let $Φ_{1} U = (\begin{matrix} ϕ_{11} & ϕ_{12} ϕ_{21} & ϕ_{22} \end{matrix})$ with $ϕ_{11} \in C^{n \times n}$ . Using the GSVD of ${A, B}$ in (1.1) and (1.2) we have

		$max Φ_{1} \in U_{m} t r (A^{H} Φ_{1}^{H} Q_{i} Φ_{1} A (A^{H} A + B^{H} B)^{- 1})$
	$=$	$max Φ_{1} \in U_{m} t r ((A^{H} A + B^{H} B)^{- 1 / 2} A^{H} Φ_{1}^{H} Q_{i} Φ_{1} A (A^{H} A + B^{H} B)^{- 1 / 2})$
	$=$	$max Φ_{1} \in U_{m} t r (Σ_{A}^{H} U^{H} Φ_{1}^{H} Q_{i} Φ_{1} U Σ_{A})$
	$=$	$max ϕ_{11} ϕ_{11}^{H} \leq I_{n} t r ({^Σ}_{A}^{H} ϕ_{11}^{H} T_{i} ϕ_{11} {^Σ}_{A}) .$

Observe $f (ϕ_{11}) = t r ({^Σ}_{A}^{H} ϕ_{11}^{H} T_{i} ϕ_{11} {^Σ}_{A})$ is an analytical function of several complex variables on the domain $ϕ_{11} ϕ_{11}^{H} \leq I_{n}$ . By Lemma 2.2 we have $f (ϕ_{11})$ attains its maximum modulus on the characteristic manifold ${ϕ_{11} \in C^{n \times n} : ϕ_{11} ϕ_{11}^{H} = I_{n}}$ . Therefore,

max ϕ_{11} ϕ_{11}^{H} \leq I_{n} t r ({^Σ}_{A}^{H} ϕ_{11}^{H} T_{i} ϕ_{11} {^Σ}_{A}) = max ϕ_{11} ϕ_{11}^{H} = I_{n} t r ({^Σ}_{A}^{H} ϕ_{11}^{H} T_{i} ϕ_{11} {^Σ}_{A}) .

Thus, by Lemma 2.1 we have

max Φ_{1} \in U_{m} t r (A^{H} Φ_{1}^{H} Q_{i} Φ_{1} A (A^{H} A + B^{H} B)^{- 1}) = max ϕ_{11} ϕ_{11}^{H} = I_{n} t r (^Σ_{A}^{H} ϕ_{11}^{H} T_{i} ϕ_{11} {^Σ}_{A}) = α_{1}^{2} + \dots + α_{i}^{2} .

(2.9)

Similarly, we have

max Φ_{1} \in U_{m} t r (A^{H} Φ_{1}^{H} Q_{i - 1} Φ_{1} A (A^{H} A + B^{H} B)^{- 1}) = α_{1}^{2} + \dots + α_{i - 1}^{2} .

(2.10)

From (2.9) and (2.10) it follows that (2.1) holds for $1 \leq i \leq n$ .

b) If $n \leq p$ , then for $1 \leq i \leq n$ , let $P_{i}$ be defined by (2.4). Let $Σ_{B} = (O_{(p - n) \times n}, {^Σ}_{B})$ . For any $Φ_{2} \in U_{p}$ , let $Φ_{2} V = (\begin{matrix} ψ_{11} & ψ_{12} ψ_{21} & ψ_{22} \end{matrix}) \in U_{p}$ with $ψ_{22} \in C^{n \times n}$ . Using the GSVD of ${A, B}$ in (1.1) and (1.2) we have

		$max Φ_{2} \in U_{p} t r (B^{H} Φ_{2}^{H} P_{i} Φ_{2} B (A^{H} A + B^{H} B)^{- 1})$
		$max Φ_{2} \in U_{p} t r ((A^{H} A + B^{H} B)^{- 1 / 2} B^{H} Φ_{2}^{H} P_{i} Φ_{2} B (A^{H} A + B^{H} B)^{- 1 / 2})$
	$=$	$max Φ_{2} \in U_{p} t r (Σ_{B}^{H} V^{H} Φ_{2}^{H} P_{i} Φ_{2} V Σ_{B})$
	$=$	$max ψ_{22} ψ_{22}^{H} \leq I_{n} t r ({^Σ}_{B}^{H} ψ_{22}^{H} F_{i} ψ_{22} {^Σ}_{B}),$

which attains its maximum modulus on the characteristic manifold ${ψ_{22} \in C^{n \times n} : ψ_{22} ψ_{22}^{H} = I_{n}}$ by Lemma 2.2. Then by Lemma 2.1 we have

max Φ_{2} \in U_{p} t r (B^{H} Φ_{2}^{H} P_{i} Φ_{2} B (A^{H} A + B^{H} B)^{- 1}) = max ψ_{22} ψ_{22}^{H} = I_{n} t r (^Σ_{B}^{H} ψ_{22}^{H} F_{i} ψ_{22} {^Σ}_{B}) = β_{i}^{2} + \dots + β_{n}^{2} .

(2.11)

Similarly, we have

max Ψ_{1} \in U_{p} t r (B^{H} Ψ_{1}^{H} P_{i + 1} Ψ_{1} B (A^{H} A + B^{H} B)^{- 1}) = β_{i + 1}^{2} + \dots + β_{n}^{2} .

(2.12)

Using (2.11) and (2.12) we can conclude that (2.3) holds for $1 \leq i \leq n$ .

c) If $m \leq n$ , then it is easy to check that $α_{m + 1} = \dots = α_{n} = 0$ . For $1 \leq i \leq m$ , let $Q_{i}$ be defined by (2.6). We set $Σ_{A} = ({^Σ}_{A}, O_{m \times (n - m)})$ . Using the GSVD of ${A, B}$ in (1.1) and (1.2) and Lemma 2.1 we have

	$max Ψ_{1} \in U_{m} t r (A^{H} Ψ_{1}^{H} Q_{i} Ψ_{1} A (A^{H} A + B^{H} B)^{- 1})$
$=$	$max Ψ_{1} \in U_{m} t r (Σ_{A}^{H} U^{H} Ψ_{1}^{H} Q_{i} Ψ_{1} U Σ_{A})$
$=$	$max Ψ_{1} \in U_{m} t r (d i a g ({^Σ}_{A}^{H} U^{H} Ψ_{1}^{H} Q_{i} Ψ_{1} U {^Σ}_{A}, O_{(n - m) \times (n - m)}))$
$=$	$max Ψ_{1} \in U_{m} t r ({^Σ}_{A}^{H} U^{H} Ψ_{1}^{H} Q_{i} Ψ_{1} U {^Σ}_{A})$
$=$	$α_{1}^{2} + \dots + α_{i}^{2} .$	(2.13)

Similarly, we have

max Ψ_{1} \in U_{m} t r (A^{H} Ψ_{1}^{H} Q_{i - 1} Ψ_{1} A (A^{H} A + B^{H} B)^{- 1}) = α_{1}^{2} + \dots + α_{i - 1}^{2} .

(2.14)

From (2) and (2.14) it follows that (2.5) holds for $1 \leq i \leq m$ .

d) If $p \leq n$ , then $β_{1} = \dots = β_{n - p} = 0$ . For $n - p + 1 \leq i \leq n$ , let $P_{i}$ be defined by (2.8). We set $Σ_{B} = (O_{p \times (n - p)}, {^Σ}_{B})$ . From the GSVD of ${A, B}$ in (1.1) and (1.2) and Lemma 2.1 we have

	$max Ψ_{2} \in U_{p} t r (B^{H} Ψ_{2}^{H} P_{i} Ψ_{2} B (A^{H} A + B^{H} B)^{- 1})$
$=$	$max Ψ_{2} \in U_{p} t r ((A^{H} A + B^{H} B)^{- 1 / 2} B^{H} Ψ_{2}^{H} P_{i} Ψ_{2} B (A^{H} A + B^{H} B)^{- 1 / 2})$
$=$	$max Ψ_{2} \in U_{p} t r (Σ_{B}^{H} V^{H} Ψ_{2}^{H} P_{i} Ψ_{2} V Σ_{B})$
$=$	$max Ψ_{2} \in U_{p} t r (d i a g (O_{(n - p) \times (n - p)}, {^Σ}_{B}^{H} V^{H} Ψ_{2}^{H} P_{i} Ψ_{2} V {^Σ}_{B}))$
$=$	$max Ψ_{2} \in U_{p} t r ({^Σ}_{B}^{H} V^{H} Ψ_{2}^{H} P_{i} Ψ_{2} V {^Σ}_{B})$
$=$	$β_{i}^{2} + \dots + β_{n}^{2} .$	(2.15)

Similarly, we have

max Ψ_{2} \in U_{p} t r (B^{H} Ψ_{2}^{H} P_{i + 1} Ψ_{2} B (A^{H} A + B^{H} B)^{- 1}) = β_{i + 1}^{2} + \dots + β_{n}^{2} .

(2.16)

Using (2) and (2.16) we have (2.7) holds for $n - p + 1 \leq i \leq n$ . The proof is complete.

Next, we give new formula model of any GSV of a GMP by trace function optimization under two variables.

Lemma 2.4

Let ${A, B}$ be an (m,p,n)-GMP and the GSV $(α_{i}, β_{i})$ of ${A, B}$ be given in (1.1) and (1.2). The following conclusions hold true.

If $n \leq m,$ then for $1 \leq i \leq n$ ,

$α_{i}$ $=$ $max Π_{1} \in U_{m}, Π_{2} \in U_{n} | t r (Π_{1} A (A^{H} A + B^{H} B)^{- 1 / 2} Π_{2} G_{i}) |$ (2.17)

$- max Π_{1} \in U_{m}, Π_{2} \in U_{n} | t r (Π_{1} A (A^{H} A + B^{H} B)^{- 1 / 2} Π_{2} G_{i - 1}) | \equiv {~ ϕ}_{i},$

where

$G_{i} = (G_{i}, O_{n \times (m - n)}), G_{i} = d i a g (I_{i}, O_{(n - i) \times (n - i)}) .$ (2.18)
If $n \leq p$ , then for $1 \leq i \leq n$ ,

$β_{i}$ $=$ $max Ξ_{1} \in U_{p}, Ξ_{2} \in U_{n} | t r (Ξ_{1} B (A^{H} A + B^{H} B)^{- 1 / 2} Ξ_{2} H_{i}) |$ (2.19)

$- max Ξ_{1} \in U_{p}, Ξ_{2} \in U_{n} | t r (Ξ_{1} B (A^{H} A + B^{H} B)^{- 1 / 2} Ξ_{2} H_{i - 1}) | \equiv {~ ψ}_{i},$

where

$H_{i} = (O_{n \times (p - n)}, H_{i}), H_{i} = d i a g (O_{(i - 1) \times (i - 1)}, I_{n - i + 1}) .$ (2.20)
If $m \leq n$ , then $α_{m + 1} = \dots = α_{n} = 0$ and for $1 \leq i \leq m$ ,

$α_{i}$ $=$ $max Π_{3} \in U_{n}, Π_{4} \in U_{m} | t r (Π_{3} (A^{H} A + B^{H} B)^{- 1 / 2} A^{H} Π_{4} S_{i}) |$ (2.21)

$- max Π_{3} \in U_{n}, Π_{4} \in U_{m} | t r (Π_{3} (A^{H} A + B^{H} B)^{- 1 / 2} A^{H} Π_{4} S_{i - 1}) | \equiv {~ φ}_{i},$

$S_{i} = (S_{i}, O_{m \times (n - m)}), S_{i} = d i a g (I_{i}, O_{(m - i) \times (m - i)}) .$ (2.22)
If $p \leq n$ , then $β_{1} = \dots = β_{n - p} = 0$ and for $n - p + 1 \leq i \leq n$ ,

$β_{i}$ $=$ $max Ξ_{3} \in U_{n}, Ξ_{4} \in U_{p} | t r (Ξ_{3} (A^{H} A + B^{H} B)^{- 1 / 2} B^{H} Ξ_{4} W_{i}) |$ (2.23)

$- max Ξ_{3} \in U_{n}, Ξ_{4} \in U_{p} | t r (Ξ_{3} (A^{H} A + B^{H} B)^{- 1 / 2} B^{H} Ξ_{4} W_{i - 1}) | \equiv {~ χ}_{i},$

where

$W_{i} = (O_{p \times (n - p)}, W_{i}), W_{i} = d i a g (O_{(p - n + i - 1) \times (p - n + i - 1)}, I_{n - i + 1}) .$ (2.24)

Proof: a) If $n \leq m,$ then for $1 \leq i \leq n$ , we let $G_{i}$ be defined by (2.18). We set $Σ_{A} = (\begin{matrix} {^Σ}_{A} O_{(m - n) \times n} \end{matrix})$ . For any $Π_{1} \in U_{m}$ , let $Π_{1} U = (\begin{matrix} Π_{11} & Π_{12} Π_{21} & Π_{22} \end{matrix})$ with $Π_{11} \in C^{n \times n}$ . By the GSVD of ${A, B}$ in (1.1) and (1.2) we have

		$max Π_{1} \in U_{m}, Π_{2} \in U_{n} \| t r (Π_{1} A (A^{H} A + B^{H} B)^{- 1 / 2} Π_{2} G_{i}) \|$
	$=$	$max Π_{1} \in U_{m}, Π_{2} \in U_{n} \| t r (Π_{1} U Σ_{A} R (R^{H} R)^{- 1 / 2} Π_{2} G_{i}) \|$
	$=$	$max Π_{11} Π_{11}^{H} \leq I_{n}, Π_{2} \in U_{n} \| t r (Π_{11} {^Σ}_{A} R (R^{H} R)^{- 1 / 2} Π_{2} G_{i}) \| .$

Observe $f (Π_{11}) = t r (Π_{11} {^Σ}_{A} R (R^{H} R)^{- 1 / 2} Π_{2} G_{i})$ is an analytical function of several complex variables on the domain $Π_{11} Π_{11}^{H} \leq I_{n}$ . By Lemma 2.2 we know that $f (Π_{11})$ attains its maximum modulus on the characteristic manifold ${Π_{11} \in C^{n \times n} : Π_{11} Π_{11}^{H} = I_{n}}$ . Therefore,

			$max Π_{11} Π_{11}^{H} \leq I_{n}, Π_{2} \in U_{n} \| t r (Π_{11} {^Σ}_{A} R (R^{H} R)^{- 1 / 2} Π_{2} G_{i}) \|$
		$=$	$max Π_{11} Π_{11}^{H} = I_{n}, Π_{2} \in U_{n} \| t r (Π_{11} {^Σ}_{A} R (R^{H} R)^{- 1 / 2} Π_{2} G_{i}) \|$

and it follows from Lemma 2.1 that

	$max Π_{1} \in U_{m}, Π_{2} \in U_{n} \| t r (Π_{1} A (A^{H} A + B^{H} B)^{- 1 / 2} Π_{2} G_{i}) \|$
	$= max Π_{11} Π_{11}^{H} = I_{n}, Π_{2} \in U_{n} \| t r (Π_{11} {^Σ}_{A} R (R^{H} R)^{- 1 / 2} Π_{2} G_{i}) \|$
	$= α_{1} + \dots + α_{i} .$		(2.25)

Similarly, we have

max Π_{1} \in U_{m}, Π_{2} \in U_{n} | t r (Π_{1} A (A^{H} A + B^{H} B)^{- 1 / 2} Π_{2} G_{i - 1}) | = α_{1} + \dots + α_{i - 1} .

(2.26)

From (2) and (2.26) we know that (2.17) holds for $1 \leq i \leq n$ .

b) If $n \leq p$ , then for $1 \leq i \leq n$ , let $H_{i}$ be defined by (2.20). We set $Σ_{B} = (\begin{matrix} O_{(p - n) \times n} {^Σ}_{B} \end{matrix})$ . For any $Ξ_{1} \in U_{p}$ , let $Ξ_{1} V = (\begin{matrix} Ξ_{11} & Ξ_{12} Ξ_{21} & Ξ_{22} \end{matrix}) \in U_{p}$ with $Ξ_{11} \in C^{n \times n}$ . By the GSVD of ${A, B}$ in (1.1) and (1.2), Lemma 2.2, and Lemma 2.1 we have

max Ξ_{1} \in U_{p}, Ξ_{2} \in U_{n} | t r (Ξ_{1} B (A^{H} A + B^{H} B)^{- 1}

New Formulation and Computation for Generalized Singular Values of Grassman Matrix Pair

Authors

On choices of formulations of computing the generalized singular value decomposition of a matrix pair

Cross product-free matrix pencils for computing generalized singular values

Two harmonic Jacobi–Davidson methods for computing a partial generalized singular value decomposition of a large matrix pair

Takagi factorization of matrices depending on parameters and locating degeneracies of singular values

Singularity-Avoiding Multi-Dimensional Root-Finder

A rounding error analysis of the joint bidiagonalization process with applications to the GSVD computation

A Study of the Complexity and Accuracy of Direction of Arrival Estimation Methods Based on GCC-PHAT for a Pair of Close Microphones

1 Introduction

1.1 Organization

1.2 Notation

Definition 1.1

Definition 1.2

Definition 1.3

2 Trace function optimization model formulation

Lemma 2.1

Lemma 2.2

Lemma 2.3

Lemma 2.4

	$α_{i}^{2}$	$=$	$max Φ_{1} \in U_{m} t r (A^{H} Φ_{1}^{H} Q_{i} Φ_{1} A (A^{H} A + B^{H} B)^{- 1})$		(2.1)
			$- max Φ_{1} \in U_{m} t r (A^{H} Φ_{1}^{H} Q_{i - 1} Φ_{1} A (A^{H} A + B^{H} B)^{- 1}) \equiv ϕ_{i},$		(2.1)

	$β_{i}^{2}$	$=$	$max Φ_{2} \in U_{p} t r (B^{H} Φ_{2}^{H} P_{i} Φ_{2} B (A^{H} A + B^{H} B)^{- 1})$		(2.3)
			$- max Φ_{2} \in U_{p} t r (B^{H} Φ_{2}^{H} P_{i - 1} Φ_{1} B (A^{H} A + B^{H} B)^{- 1}) \equiv ψ_{i},$		(2.3)

	$α_{i}^{2}$	$=$	$max Ψ_{1} \in U_{m} t r (A^{H} Ψ_{1}^{H} Q_{i} Ψ_{1} A (A^{H} A + B^{H} B)^{- 1})$		(2.5)
			$- max Ψ_{1} \in U_{m} t r (A^{H} Ψ_{1}^{H} Q_{i - 1} Ψ_{1} A (A^{H} A + B^{H} B)^{- 1}) \equiv φ_{i},$		(2.5)

	$β_{i}^{2}$	$=$	$max Ψ_{2} \in U_{p} t r (B^{H} Ψ_{2}^{H} P_{i} Ψ_{2} B (A^{H} A + B^{H} B)^{- 1})$		(2.7)
			$- max Ψ_{2} \in U_{p} t r (B^{H} Ψ_{2}^{H} P_{i - 1} Ψ_{2} B (A^{H} A + B^{H} B)^{- 1}) \equiv χ_{i} .$		(2.7)

	$α_{i}$	$=$	$max Π_{1} \in U_{m}, Π_{2} \in U_{n} \| t r (Π_{1} A (A^{H} A + B^{H} B)^{- 1 / 2} Π_{2} G_{i}) \|$		(2.17)
			$- max Π_{1} \in U_{m}, Π_{2} \in U_{n} \| t r (Π_{1} A (A^{H} A + B^{H} B)^{- 1 / 2} Π_{2} G_{i - 1}) \| \equiv {~ ϕ}_{i},$		(2.17)

	$β_{i}$	$=$	$max Ξ_{1} \in U_{p}, Ξ_{2} \in U_{n} \| t r (Ξ_{1} B (A^{H} A + B^{H} B)^{- 1 / 2} Ξ_{2} H_{i}) \|$		(2.19)
			$- max Ξ_{1} \in U_{p}, Ξ_{2} \in U_{n} \| t r (Ξ_{1} B (A^{H} A + B^{H} B)^{- 1 / 2} Ξ_{2} H_{i - 1}) \| \equiv {~ ψ}_{i},$		(2.19)

	$α_{i}$	$=$	$max Π_{3} \in U_{n}, Π_{4} \in U_{m} \| t r (Π_{3} (A^{H} A + B^{H} B)^{- 1 / 2} A^{H} Π_{4} S_{i}) \|$		(2.21)
			$- max Π_{3} \in U_{n}, Π_{4} \in U_{m} \| t r (Π_{3} (A^{H} A + B^{H} B)^{- 1 / 2} A^{H} Π_{4} S_{i - 1}) \| \equiv {~ φ}_{i},$		(2.21)

	$β_{i}$	$=$	$max Ξ_{3} \in U_{n}, Ξ_{4} \in U_{p} \| t r (Ξ_{3} (A^{H} A + B^{H} B)^{- 1 / 2} B^{H} Ξ_{4} W_{i}) \|$		(2.23)
			$- max Ξ_{3} \in U_{n}, Ξ_{4} \in U_{p} \| t r (Ξ_{3} (A^{H} A + B^{H} B)^{- 1 / 2} B^{H} Ξ_{4} W_{i - 1}) \| \equiv {~ χ}_{i},$		(2.23)

New Formulation and Computation for Generalized Singular Values of Grassman Matrix Pair

Authors

Related Research

On choices of formulations of computing the generalized singular value decomposition of a matrix pair

Cross product-free matrix pencils for computing generalized singular values

Two harmonic Jacobi–Davidson methods for computing a partial generalized singular value decomposition of a large matrix pair

Takagi factorization of matrices depending on parameters and locating degeneracies of singular values

Singularity-Avoiding Multi-Dimensional Root-Finder

A rounding error analysis of the joint bidiagonalization process with applications to the GSVD computation

A Study of the Complexity and Accuracy of Direction of Arrival Estimation Methods Based on GCC-PHAT for a Pair of Close Microphones

This week in AI

1 Introduction

1.1 Organization

1.2 Notation

Definition 1.1

Definition 1.2

Definition 1.3

2 Trace function optimization model formulation

Lemma 2.1

Lemma 2.2

Lemma 2.3

Lemma 2.4