Chapter 6Matrices and Indefinite ScalarProducts

Having introduced main notions of the geometry based on an indefinite scalarproduct we will now study special classes of matrices intimately connectedwith this scalar product. These will be the analogs of the usual Hermitian andunitary matrices. At first sight the formal analogy seems complete but theindefiniteness of the underlying scalar product often leads to big, sometimessurprising, differences.

A matrix 𝐻 ∈ 𝛯𝑛,𝑛 is called 𝐽-Hermitian (also 𝐽-symmetric, if real), if

𝐻∗ = JHJ ⇔ (JH)∗ = JH

or, equivalently,

[𝐻𝑥, 𝑦] = 𝑦∗JH𝑥 = 𝑦∗𝐻∗𝐽𝑥 = [𝑥,Hy].

It is convenient to introduce the 𝐽-adjoint 𝐴[∗] or the 𝐽-transpose 𝐴[𝑇 ] of ageneral matrix 𝐴, defined as

𝐴[∗] = JA∗𝐽, 𝐴[𝑇 ] = JA𝑇𝐽,

respectively. In the latter case the symmetry 𝐽 is supposed to be real. Nowthe 𝐽-Hermitian property is characterised by

𝐴[∗] = 𝐴

or, equivalently by

[𝐴𝑥, 𝑦] = [𝑥,𝐴𝑦] for all 𝑥, 𝑦 ∈ 𝛯𝑛.

A matrix 𝑈 ∈ 𝛯𝑛,𝑛 is 𝐽-unitary (also 𝐽-orthogonal, if real), if

𝑈−1 = 𝑈 [∗] = 𝐽𝑈∗𝐽 ⇔ 𝑈∗𝐽𝑈 = 𝐽.

K. Veselic, Damped Oscillations of Linear Systems,Lecture Notes in Mathematics 2023, DOI 10.1007/978-3-642-21335-9 6,© Springer-Verlag Berlin Heidelberg 2011

49

50 6 Matrices and Indefinite Scalar Products

Obviously all 𝐽-unitaries form a multiplicative group and satisfy

∣ det𝑈 ∣ = 1.

The 𝐽-unitarity can be expressed by the identity

[𝑈𝑥,𝑈𝑦] = 𝑦∗𝑈∗𝐽𝑈𝑥 = [𝑥, 𝑦].

Exercise 6.1 Prove𝐼 [∗] = 𝐼

(𝛼𝐴 + 𝛽𝐵)[∗] = 𝛼𝐴[∗] + 𝛽𝐵[∗]

(𝐴𝐵)[∗] = 𝐵[∗]𝐴[∗]

(𝐴[∗])−1 = (𝐴−1)[∗]

𝐴[∗] = 𝐴−1 ⇐⇒ A is 𝐽-unitary

In the particular case 𝐽 =

[𝐼 0

0 −𝐼

]a 𝐽-Hermitian 𝐴 looks like

𝐴 =

[𝐴11 𝐴12

−𝐴∗12 𝐴22

], 𝐴∗

11 = 𝐴11, 𝐴∗22 = 𝐴22

whereas for 𝐽 =

[0 𝐼

𝐼 0

]the 𝐽-Hermitian is

𝐴 =

[𝐴11 𝐴12

𝐴21 𝐴∗11

], 𝐴∗

12 = 𝐴12, 𝐴∗21 = 𝐴21. (6.1)

By the unitary invariance of the spectral norm the condition number of a𝐽-unitary matrix 𝑈 is

𝜅(𝑈) = ∥𝑈∥∥𝑈−1∥ = ∥𝑈∥∥𝐽𝑈∗𝐽∥ = ∥𝑈∥∥𝑈∗∥ = ∥𝑈∥2 ≥ 1

We call a matrix jointly unitary, if it is simultaneously 𝐽-unitary and unitary.Examples of jointly unitary matrices 𝑈 are given in (3.11) and (4.24).

Exercise 6.2 Prove that the following are equivalent

(i) U is jointly unitary of order 𝑛.(ii) 𝑈 is 𝐽-unitary and ∥𝑈∥ = 1.(iii) 𝑈 is 𝐽-unitary and 𝑈 commutes with 𝐽 .(iv) 𝑈 is unitary and 𝑈 commutes with 𝐽 .(v) 𝑈 is 𝐽-unitary and ∥𝑈∥2𝐸 = Tr𝑈∗𝑈 = 𝑛.

6 Matrices and Indefinite Scalar Products 51

Example 6.3 Any matrix of the form

𝑌 = 𝐻 (𝑊 )

(𝑉1 0

0 𝑉2

), 𝐻 (𝑊 ) =

(√𝐼 + 𝑊𝑊 ∗ 𝑊

𝑊 ∗ √𝐼 + 𝑊 ∗𝑊

)(6.2)

is obviously 𝐽-unitary with 𝐽 from (5.6); here 𝑊 is an 𝑚 × (𝑛−𝑚)-matrixand 𝑉1, 𝑉2 are unitary. As a matter of fact, any 𝐽-unitary 𝑈 is of this form.This we will now show. Any 𝐽-unitary 𝑈 , partitioned according to (5.6) iswritten as

𝑈 =

[𝑈11 𝑈12

𝑈21 𝑈22

].

By taking the polar decompositions 𝑈11 = 𝑊11𝑉1, 𝑈22 = 𝑊22𝑉2 with 𝑊11 =√𝑈11𝑈∗

11, 𝑊22 =√𝑈22𝑈∗

22 we have

𝑈 =

[𝑊11 𝑊12

𝑊21 𝑊22

] [𝑉1 0

0 𝑉2

].

Now in the product above the second factor is 𝐽-unitary (𝑉1,2 being unitary).Thus, the first factor – we call it 𝐻 – is also 𝐽-unitary, that is, 𝐻∗𝐽𝐻 = 𝐽or, equivalently, 𝐻𝐽𝐻∗ = 𝐽 . This is expressed as

𝑊 211 −𝑊 ∗

21𝑊21 = 𝐼𝑚 𝑊 211 −𝑊12𝑊

∗12 = 𝐼𝑚

𝑊11𝑊12 −𝑊 ∗21𝑊22 = 0 𝑊11𝑊

∗21 −𝑊12𝑊22 = 0

𝑊 ∗12𝑊12 −𝑊 2

22 = −𝐼𝑛−𝑚 𝑊21𝑊∗21 −𝑊 2

22 = −𝐼𝑛−𝑚

This gives (note that 𝑊11,𝑊22 are Hermitian positive semidefinite)

√𝐼𝑚 + 𝑊 ∗

21𝑊21𝑊12 = 𝑊 ∗21

√𝐼𝑛−𝑚 + 𝑊21𝑊 ∗

21

or, equivalently,

𝑊12(𝐼𝑛−𝑚 + 𝑊 ∗12𝑊12)

−1/2 = (𝐼𝑚 + 𝑊 ∗21𝑊21)

−1/2𝑊 ∗21

= 𝑊 ∗21(𝐼𝑛−𝑚 + 𝑊21𝑊

∗21)

−1/2 = 𝑊 ∗21(𝐼𝑛−𝑚 + 𝑊 ∗

12𝑊12)−1/2

hence 𝑊 ∗21 = 𝑊12. Here the second equality follows from the general identity

𝐴𝑓(𝐵𝐴) = 𝑓(𝐴𝐵)𝐴 which we now assume as known and will address indiscussing analytic matrix functions later. Now set 𝑊 = 𝑊12 and obtain(6.2).

Jointly unitary matrices are very precious whenever they can be used incomputations, because their condition is equal to one.

If 𝐴 is 𝐽-Hermitian and 𝑈 is 𝐽-unitary then one immediately verifies that

𝐴′ = 𝑈−1𝐴𝑈 = 𝑈 [∗]𝐴𝑈

is again 𝐽-Hermitian.

52 6 Matrices and Indefinite Scalar Products

If 𝐽 is diagonal then the columns (and also the rows) of any 𝐽-unitarymatrix form a 𝐽-orthonormal basis. It might seem odd that the converse isnot true. This is so because in our definition of 𝐽-orthonormality the orderof the vectors plays no role. For instance the vectors

⎡⎣ 1

0

0

⎤⎦ ,

⎡⎣ 0

0

1

⎤⎦ ,

⎡⎣ 0

1

0

⎤⎦

are 𝐽-orthonormal with respect to

𝐽 =

⎡⎣ 1

−1

1

⎤⎦

but the matrix 𝑈 built from these three vectors in that order is not 𝐽-orthogonal. We rather have

𝑈∗𝐽𝑈 = 𝐽 ′ =

⎡⎣ 1

1

−1

⎤⎦ .

To overcome such difficulties we call a square matrix 𝐽, 𝐽 ′-unitary (𝐽, 𝐽 ′-orthogonal, if real), if

𝑈∗𝐽𝑈 = 𝐽 ′, (6.3)

where 𝐽 and 𝐽 ′ are symmetries. If in (6.3) the orders of 𝐽 ′ and 𝐽 do notcoincide we call 𝑈 a 𝐽, 𝐽 ′-isometry. If both 𝐽 and 𝐽 ′ are unit matrices thisis just a standard isometry.

Exercise 6.4 Any 𝐽, 𝐽 ′- isometry is injective.

Proposition 6.5 If 𝐽, 𝐽 ′ are symmetries and 𝑈 is 𝐽, 𝐽 ′-unitary then 𝐽 and𝐽 ′ are unitarily similar:

𝐽 ′ = 𝑉 −1𝐽𝑉, 𝑉 unitary (6.4)

and𝑈 = 𝑊𝑉, (6.5)

where 𝑊 is 𝐽-unitary.

Proof. The eigenvalues of both 𝐽 and 𝐽 ′ consists of ± ones. By (6.3) 𝑈 mustbe non-singular but then (6.3) and the theorem of Sylvester imply that 𝐽 and𝐽 ′ have the same eigenvalues including multiplicities, so they are unitarilysimilar. Now (6.5) follows from (6.4). Q.E.D.

6 Matrices and Indefinite Scalar Products 53

If 𝐴 is 𝐽-Hermitian and 𝑈 is 𝐽, 𝐽 ′-unitary then

𝐴′ = 𝑈−1𝐴𝑈

is 𝐽 ′-Hermitian. Indeed,

𝐴′∗ = 𝑈∗𝐴∗𝑈−∗ = 𝑈∗AJUJ′.

Here 𝐴𝐽 , and therefore 𝑈∗𝐴𝐽𝑈 is Hermitian so 𝐴′∗ is 𝐽 ′-Hermitian.Using only 𝐽-unitary similarities the 𝐽-Hermitian matrix

𝐴 =

⎡⎣ 1 0 −5

0 2 0

−5 0 1

⎤⎦ , 𝐽 =

⎡⎣ 1

1

−1

⎤⎦ (6.6)

cannot be further simplified, but using the 𝐽, 𝐽 ′-unitary matrix

𝛱 =

⎡⎣ 1 0 0

0 0 1

0 1 0

⎤⎦ , 𝐽 ′ =

⎡⎣1

−1

1

⎤⎦

we obtain the more convenient block-diagonal form

𝛱−1𝐴𝛱 =

⎡⎣ 1 −5 0

−5 1 0

0 0 2

⎤⎦

which may have computational advantages. Since any 𝐽, 𝐽 ′-unitary matrix 𝑈is a product of a unitary matrix and a 𝐽-unitary matrix we have

𝜅(𝑈) = ∥𝑈∥2 ≥ 1,

where the equality is attained, if and only if 𝑈 is unitary.Mapping by a 𝐽, 𝐽 ′-unitary matrix 𝑈 preserves the corresponding ‘indefi-

nite geometries’ e.g.

• If 𝑥′ = 𝑈𝑥, 𝑦′ = 𝑈𝑦 then 𝑥∗𝐽𝑦 = 𝑥′∗𝐽 ′𝑦′, 𝑥∗𝐽𝑥 = 𝑥′∗𝐽 ′𝑥′

• If 𝒳 is a subspace and 𝒳 ′ = 𝑈𝒳 then 𝒳 ′ is 𝐽 ′-non-degenerate, if and onlyif 𝒳 is 𝐽-non-degenerate (the same with ‘positive’, ‘non-negative’, ‘neutral’etc.)

• The inertia does not change:

𝜄±(𝒳 ) = 𝜄′±(𝒳 ′),

where 𝜄′± is related to 𝐽 ′.

54 6 Matrices and Indefinite Scalar Products

The set of 𝐽, 𝐽 ′-unitary matrices is not essentially larger than theone of standard 𝐽-unitaries but it is often more convenient in numericalcomputations.

Exercise 6.6 Find all real 𝐽-orthogonals and all complex 𝐽-unitaries oforder 2 with

𝐽 =

[1 0

0 −1

]or 𝐽 =

[0 1

1 0

].

Exercise 6.7 A matrix is called jointly Hermitian, if it is both Hermitianand 𝐽-Hermitian. Prove that the following are equivalent

(i) H is jointly Hermitian.(ii) 𝐻 is 𝐽-Hermitian and it commutes with 𝐽 .(iii) 𝐻 is Hermitian and it commutes with 𝐽 .