Schaum’s Outline Linear Algebra, Sixth Edition

Bilinear, Quadratic, and Hermitian Forms

12.1 Introduction

This chapter generalizes the notions of linear mappings and linear functionals. Specifically, we introduce the notion of a bilinear form. These bilinear maps also give rise to quadratic and Hermitian forms. Although quadratic forms were discussed previously, this chapter is treated independently of the previous results.

Although the field K is arbitrary, we will later specialize to the cases K = R and K = C. Furthermore, we may sometimes need to divide by 2. In such cases, we must assume that 1 + 1 ≠ 0, which is true when K = R or K = C.

12.2 Bilinear Forms

Let V be a vector space of finite dimension over a field K. A bilinear form on V is a mapping f:V × V → K such that, for all a, b ∈ K and all u_i, ν_i ∈ V:

(i) f(au₁ + bu₂, v) = af(u₁, v) + bf(u₂, v),

(ii) f (u; av₁ + bv₂) = af (u; v₁)+ bf (u; v₂)

We express condition (i) by saying f is linear in the first variable, and condition (ii) by saying f is linear in the second variable.

EXAMPLE 12.1

(a) Let f be the dot product on Rⁿ; that is, for u = (a_i) and v = (b_i),

Then f is a bilinear form on Rⁿ. (In fact, any inner product on a real vector space V is a bilinear form on V.)

(b) Let ϕ and σ be arbitrarily linear functionals on V. Let f:V × V → K be defined by f(u; v) = ϕ(u)σ(v). Then f is a bilinear form, because f and σ are each linear.

(c) Let A = [a_ij] be any n × n matrix over a field K. Then A may be identified with the following bilinear form F on Kⁿ, where X = [x]_i and Y = [y]_i are column vectors of variables:

The above formal expression in the variables x_i, y_i is termed the bilinear polynomial corresponding to the matrix A. Equation (12.1) shows that, in a certain sense, every bilinear form is of this type.

Space of Bilinear Forms

Let B(V) denote the set of all bilinear forms on V. A vector space structure is placed on B(V), where for any f, g ∈ B(V) and any k ∈ K, we define f + g and kf as follows:

The following theorem (proved in Problem 12.4) applies.

THEOREM 12.1: Let V be a vector space of dimension n over K. Let {ϕ₁, … ; ϕ_n} be any basis of the dual space V*. Then { f_ij : i; j = 1; … ; n} is a basis of B(V), where f_ij is defined by f_ij(u; v) = ϕ_i(u)ϕ_j(v). Thus, in particular, dim B(V) = n².

12.3 Bilinear Forms and Matrices

Let f be a bilinear form on V and let S = {u₁, … ; u_n} be a basis of V. Suppose u; v ∈ V and

Then

Thus, f is completely determined by the n² values f(u_i, u_j).

The matrix A = [a_ij] where a_ij = f(u_i, u_j) is called the matrix representation of f relative to the basis S or, simply, the “matrix of f in S.” It “represents” f in the sense that, for all u, v ∈ V,

[As usual, [u]_s denotes the coordinate (column) vector of u in the basis S.]

Change of Basis, Congruent Matrices

We now ask, how does a matrix representing a bilinear form transform when a new basis is selected? The answer is given in the following theorem (proved in Problem 12.5).

THEOREM 12.2: Let P be a change-of-basis matrix from one basis S to another basis S⁰. If A is the matrix representing a bilinear form f in the original basis S, then B = P^TAP is the matrix representing f in the new basis S⁰.

The above theorem motivates the following definition.

DEFINITION: A matrix B is congruent to a matrix A, written B ≃ A, if there exists a nonsingular matrix P such that B = P^TAP.

Thus, by Theorem 12.2, matrices representing the same bilinear form are congruent. We remark that congruent matrices have the same rank, because P and P^T are nonsingular; hence, the following definition is well defined.

DEFINITION: The rank of a bilinear form f on V, written rank(f), is the rank of any matrix representation of f. We say f is degenerate or nondegenerate according to whether rank(f) < dim V or rank(f) = dim V.

12.4 Alternating Bilinear Forms

Let f be a bilinear form on V. Then f is called

(i) alternating if f(v, v) = 0 for every v ∈ V;

(ii) skew-symmetric if f(u, v) = –f(v, u) for every u, v ∈ V.

Now suppose (i) is true. Then (ii) is true, because, for any u; v; ∈ V,

On the other hand, suppose (ii) is true and also 1 + 1 ≠ 0. Then (i) is true, because, for every v ∈ V, we have f(v, v) = f–(v, v). In other words, alternating and skew-symmetric are equivalent when 1 + 1 ≠ 0.

The main structure theorem of alternating bilinear forms (proved in Problem 12.23) is as follows.

THEOREM 12.3: Let f be an alternating bilinear form on V. Then there exists a basis of V in which f is represented by a block diagonal matrix M of the form

Moreover, the number of nonzero blocks is uniquely determined by f [because it is equal to rank(f).

In particular, the above theorem shows that any alternating bilinear form must have even rank.

12.5 Symmetric Bilinear Forms, Quadratic Forms

This section investigates the important notions of symmetric bilinear forms and quadratic forms and their representation by means of symmetric matrices. The only restriction on the field K is that 1 + 1 ≠ 0. In Section 12.6, we will restrict K to be the real field R, which yields important special results.

Symmetric Bilinear Forms

Let f be a bilinear form on V. Then f is said to be symmetric if, for every u, v ∈ V,

f(u, v) = f(v, u)

One can easily show that f is symmetric if and only if any matrix representation A of f is a symmetric matrix.

The main result for symmetric bilinear forms (proved in Problem 12.10) is as follows. (We emphasize that we are assuming that 1 + 1 ≠ 0.)

THEOREM 12.4: Let f be a symmetric bilinear form on V. Then V has a basis {v₁, … ; v_n} in which f is represented by a diagonal matrix—that is, where f(v_i, v_j) = 0 for i ≠ j.

THEOREM 12.4: (Alternative Form) Let A be a symmetric matrix over K. Then A is congruent to a diagonal matrix; that is, there exists a nonsingular matrix P such that P^TAP is diagonal.

Diagonalization Algorithm

Recall that a nonsingular matrix P is a product of elementary matrices. Accordingly, one way of obtaining the diagonal form D = P^TAP is by a sequence of elementary row operations and the same sequence of elementary column operations. This same sequence of elementary row operations on the identity matrix I will yield P^T. This algorithm is formalized below.

ALGORITHM 12.1: (Congruence Diagonalization of a Symmetric Matrix) The input is a symmetric matrix A = [a_ij] of order n.

Step 1. Form the n × 2n (block) matrix M = [A₁, I], where A₁ = A is the left half of M and the identity matrix I is the right half of M.

Step 2. Examine the entry a₁₁. There are three cases.

Case I: a₁₁ ≠ 0. (Use a₁₁ as a pivot to put 0’s below a₁₁ in M and to the right of a₁₁ in A₁:) For i = 2; … ; n:

(a) Apply the row operation “Replace R_i by –a_i₁R₁ + a₁₁R_i.”

(b) Apply the corresponding column operation “Replace C_i by a_i₁C₁ + a₁₁C_i.”

These operations reduce the matrix M to the form

Case II: a₁₁ = 0 but a_kk ≠ 0, for some k > 1.

(a) Apply the row operation “Interchange R₁ and R_k.”

(b) Apply the corresponding column operation “Interchange C₁ and C_k.”

(These operations bring a_kk into the first diagonal position, which reduces the matrix to Case I.)

Case III: All diagonal entries a_ii = 0 but some a_ij ≠ 0.

(a) Apply the row operation “Replace R_i by R_j + R_i.”

(b) Apply the corresponding column operation “Replace C_i by C_j + C_i.”

(These operations bring 2a_ij into the ith diagonal position, which reduces the matrix to Case II.)

Thus, M is finally reduced to the form (*), where A₂ is a symmetric matrix of order less than A.

Step 3. Repeat Step ∈ with each new matrix A_k (by neglecting the first row and column of the preceding matrix) until A is diagonalized. Then M is transformed into the form M⁰ = [D, Q], where D is diagonal.

Step 4. Set P = Q^T. Then D = P^TAP.

Remark 1: We emphasize that in Step 2, the row operations will change both sides of M, but the column operations will only change the left half of M.

Remark 2: The condition 1 + 1 ≠ 0 is used in Case III, where we assume that 2a_ij ≠ 0 when a_ij ≠ 0.

The justification for the above algorithm appears in Problem 12.9.

EXAMPLE 12.2 Let Images . Apply Algorithm 9.1 to find a nonsingular matrix P such that D = P^TAP is diagonal.

First form the block matrix M = [A, I] that is, let

Apply the row operations “Replace R₂ by –2R₁ + R₂” and “Replace R₃ by 3R₁ + R₃” to M, and then apply the corresponding column operations “Replace C₂ by –2C₁ + C₂” and “Replace C₃ by 3C₁ + C₃” to obtain

Next apply the row operation “Replace R₃ by 2R₂ + R₃” and then the corresponding column operation “Replace C₃ by 2C₂ + C₃” to obtain

Now A has been diagonalized. Set

We emphasize that P is the transpose of the right half of the final matrix.

Quadratic Forms

We begin with a definition.

DEFINITION A: A mapping q:V → K is a quadratic form if q(v) = f(v, v) for some symmetric bilinear form f on V.

If 1 + 1 ≠ 0 in K, then the bilinear form f can be obtained from the quadratic form q by the following polar form of f:

Now suppose f is represented by a symmetric matrix A = [a_ij], and 1 + 1 ≠ 0. Letting X = [x]_i denote a column vector of variables, q can be represented in the form

The above formal expression in the variables x_i is also called a quadratic form. Namely, we have the following second definition.

DEFINITION B: A quadratic form q in variables x₁, x₂, … ; x_n is a polynomial such that every term has degree two; that is,

Using 1 + 1 ≠ 0, the quadratic form q in Definition B determines a symmetric matrix A = [a_ij] where . Thus, Definitions A and B are essentially the same. If the matrix representation A of q is diagonal, then q has the diagonal representation

That is, the quadratic polynomial representing q will contain no “cross product” terms. Moreover, by Theorem 12.4, every quadratic form has such a representation (when 1 + 1 ≠ 0).

12.6 Real Symmetric Bilinear Forms, Law of Inertia

This section treats symmetric bilinear forms and quadratic forms on vector spaces V over the real field R. The special nature of R permits an independent theory. The main result (proved in Problem 12.14) is as follows.

THEOREM 12.5: Let f be a symmetric form on V over R. Then there exists a basis of V in which f is represented by a diagonal matrix. Every other diagonal matrix representation of f has the same number p of positive entries and the same number n of negative entries.

The above result is sometimes called the Law of Inertia or Sylvester’s Theorem. The rank and signature of the symmetric bilinear form f are denoted and defined by

These are uniquely defined by Theorem 12.5.

A real symmetric bilinear form f is said to be

(i) positive definite if q(v) = f(v, v) > 0 for every v ≠ 0,

(ii) nonnegative semidefinite if q(v) = f(v; v) ≥ 0 for every v.

EXAMPLE 12.3 Let f be the dot product on Rⁿ. Recall that f is a symmetric bilinear form on Rⁿ. We note that f is also positive definite. That is, for any u = (a_i) ≠ 0 Rⁿ

Section 12.5 and Chapter 13 tell us how to diagonalize a real quadratic form q or, equivalently, a real symmetric matrix A by means of an orthogonal transition matrix P. If P is merely nonsingular, then q can be represented in diagonal form with only 1’s and 1’s as nonzero coefficients. Namely, we have the following corollary.

COROLLARY 12.6: Any real quadratic form q has a unique representation in the form

Images

where r = p + n is the rank of the form.

COROLLARY 12.6: (Alternative Form) Any real symmetric matrix A is congruent to the unique diagonal matrix

Images

where r = p + n is the rank of A.

12.7 Hermitian Forms

Let V be a vector space of finite dimension over the complex field C. A Hermitian form on V is a mapping f:V × V → C such that, for all a; b ∈ C and all u_i, v ∈ V,

(i) f(au₁ + bu₂, v) = af(u₁, v)+ bf (u₂, v),

(ii)

(As usual, k denotes the complex conjugate of k ∈ C.)

Using (i) and (ii), we get

That is,

As before, we express condition (i) by saying f is linear in the first variable. On the other hand, we express condition (iii) by saying f is “conjugate linear” in the second variable. Moreover, condition (ii) tells us that , and hence, f(v,v) is real for every v ∈ V.

The results of Sections 12.5 and 12.6 for symmetric forms have their analogues for Hermitian forms. Thus, the mapping q : V ℯ C, defined by q(v) = f(v, v), is called the Hermitian quadratic form or complex quadratic form associated with the Hermitian form f. We can obtain f from q by the polar form

Now suppose S = {u₁, … , u_n} is a basis of V. The matrix H = [h]_ij where h_ij = f(u_i, u_j) is called the matrix representation of f in the basis S. By (ii), Images ; hence, H is Hermitian and, in particular, the diagonal entries of H are real. Thus, any diagonal representation of f contains only real entries.

The next theorem (to be proved in Problem 12.47) is the complex analog of Theorem 12.5 on real symmetric bilinear forms.

THEOREM 12.7: Let f be a Hermitian form on V over C. Then there exists a basis of V in which f is represented by a diagonal matrix. Every other diagonal matrix representation of f has the same number p of positive entries and the same number n of negative entries.

Again the rank and signature of the Hermitian form f are denoted and defined by

These are uniquely defined by Theorem 12.7.

Analogously, a Hermitian form f is said to be

(i) positive definite if q(v) = f(v, v) > 0 for every v ≠ 0,

(ii) nonnegative semidefinite if q(v) = f(v, v) ≥ 0 for every v.

EXAMPLE 12.4 Let f be the dot product on Cⁿ; that is, for any u = (z_i) and v = (w_i) in Cⁿ,

Then f is a Hermitian form on Cⁿ. Moreover, f is also positive definite, because, for any u = (z_i) ≠ 0 in Cⁿ,

SOLVED PROBLEMS

Bilinear Forms

12.1. Let u = (x₁, x₂, x₃) and v = (y₁, y₂, y₃). Express f in matrix notation, where

Let A = [a_ij], where a_ij is the coefficient of x_iy_j. Then

12.2. Let A be an n × n matrix over K. Show that the mapping f defined by f(X; Y) = X^TAY is a bilinear form on Kⁿ.

For any a, b ∈ K and any X_i Y_i ∈ Kⁿ,

Hence, f is linear in the first variable. Also,

Hence, f is linear in the second variable, and so f is a bilinear form on Kⁿ.

12.3. Let f be the bilinear form on R² defined by

(a) Find the matrix A of f in the basis {u₁ = (1, 0); u₂ = (1, 1)}.

(b) Find the matrix B of f in the basis {v₁ = (2, 1); v₂ = (1, 1)}.

(a) Set A = [a_ij], where a_ij = f(u_i, u_j). This yields

Thus, is the matrix of f in the basis fu₁, u₂}.

(b) Set B = [b_ij] , where b_ij = f(v_i, v_j). This yields

Thus, is the matrix of f in the basis {v₁, v₂}.

and

12.4. Prove Theorem 12.1: Let V be an n-dimensional vector space over K. Let {ϕ₁, … ; ϕ_n} be any basis of the dual space V*. Then { f_ij : i, j = 1; … ; n} is a basis of B(V), where f_ij is defined by f_ij(u, v) = f_i(u)f_j(v). Thus, dim B(V) = n².

Let {u₁, … ; u_n} be the basis of V dual to {ϕ_i}. We first show that {f_ij} spans B(V). Let f ∈ B(V) and suppose f(u_i, u_j) = a_ij: We claim that f = Σi;j a_ij f_ij. It suffices to show that

Images

We have

Images

as required. Hence, {f_ij} spans B(V). Next, suppose P a_ijf_ij = 0. Then for s; t = 1; … ; n, 0 =

Images

The last step follows as above. Thus, {f_ij} is independent, and hence is a basis of B(V).

12.5. Prove Theorem 12.2. Let P be the change-of-basis matrix from a basis S to a basis S^′. Let A be the matrix representing a bilinear form in the basis S. Then B = P^TAP is the matrix representing f in the basis S⁰.

Let u; v ∈ V. Because P is the change-of-basis matrix from S to S⁰, we have P[u]_S0 = [u]_S and also P[v]_S0 = [v]_S, hence, Images . Thus,

Images

Because u and v are arbitrary elements of V, P^TAP is the matrix of f in the basis S⁰.

Symmetric Bilinear Forms, Quadratic Forms

12.6. Find the symmetric matrix that corresponds to each of the following quadratic forms:

(a) q(x; y; z) = 3x² + 4xy – y² + 8xz – 6yz + z²,

(b) q⁰(x; y; z) = 3x² + xz × 2yz, (c) q⁰⁰(x; y; z) = 2x² – 5y² – 7z²

The symmetric matrix A = [a_ij] that represents q(x₁, … ; x_n) has the diagonal entry a_ii equal to the coefficient of the square term x²_i and the nondiagonal entries a_ij and a_ji each equal to half of the coefficient of the cross-product term x_ix_j. Thus,

The third matrix A^″ is diagonal, because the quadratic form q^″ is diagonal; that is, q^″ has no cross-product terms.

12.7. Find the quadratic form q(X) that corresponds to each of the following symmetric matrices:

The quadratic form q(X) that corresponds to a symmetric matrix M is defined by q(X) = X^TMX, where X = [x]_i is the column vector of unknowns.

(a) Compute as follows:

As expected, the coefficient 5 of the square term x² and the coefficient 8 of the square term y² are the diagonal elements of A, and the coefficient 6 of the cross-product term xy is the sum of the nondiagonal elements 3 and 3 of A (or twice the nondiagonal element 3, because A is symmetric).

(b) Because B is a three-square matrix, there are three unknowns, say x; y; z or x₁, x₂, x₃. Then

Here we use the fact that the coefficients of the square terms are the respective diagonal elements 4; –6; –9 of B, and the coefficient of the cross-product term x_ix_j is the sum of the nondiagonal elements b_ij and b_ji (or twice b_ij, because b_ij = b_ji).

12.8. Let Images Apply Algorithm 12.1 to find a nonsingular matrix P such that D = P^TAP is diagonal, and find sig(A), the signature of A.

First form the block matrix M = [A, I] :

Using a₁₁ = 1 as a pivot, apply the row operations “Replace R₂ by 3R₁ + R₂” and “Replace R₃ by 2R₁ + R₃” to M and then apply the corresponding column operations “Replace C₂ by 3C₁ + C₂” and “Replace C₃ by 2C₁ + C₃” to A to obtain

Next apply the row operation “Replace R₃ by R₂ + 2R₃” and then the corresponding column operation “Replace C₃ by C₂ + 2C₃” to obtain

Now A has been diagonalized and the transpose of P is in the right half of M. Thus, set

Note D has p = ∈ positive and n = 1 negative diagonal elements. Thus, the signature of A is sig(A) = p – n = 2 ∈ 1 = 1.

12.9. Justify Algorithm 12.1, which diagonalizes (under congruence) a symmetric matrix A.

Consider the block matrix M = [A, I]. The algorithm applies a sequence of elementary row operations and the corresponding column operations to the left side of M, which is the matrix A. This is equivalent to premultiplying A by a sequence of elementary matrices, say, E₁, E₂, … ; E_r, and postmultiplying A by the transposes of the E_i. Thus, when the algorithm ends, the diagonal matrix D on the left side of M is equal to

On the other hand, the algorithm only applies the elementary row operations to the identity matrix I on the right side of M. Thus, when the algorithm ends, the matrix on the right side of M is equal to

Setting P = Q^T, we get D = P^TAP, which is a diagonalization of A under congruence.

12.10. Prove Theorem 12.4: Let f be a symmetric bilinear form on V over K (where 1 + 1 ≠ 0). Then V has a basis in which f is represented by a diagonal matrix.

Algorithm 12.1 shows that every symmetric matrix over K is congruent to a diagonal matrix. This is equivalent to the statement that f has a diagonal representation.

12.11. Let q be the quadratic form associated with the symmetric bilinear form f. Verify the polar identity Images . (Assume that 1 + 1 ≠ 0.)

We have

If 1 + 1 ≠ 0, we can divide by 2 to obtain the required identity.

12.12. Consider the quadratic form q(x, y) = 3x² + 2xy y² and the linear substitution

(a) Rewrite q(x, y) in matrix notation, and find the matrix A representing q(x, y).

(b) Rewrite the linear substitution using matrix notation, and find the matrix P corresponding to the substitution.

(d) Find q(s; t) using matrix notation.

(a) Here . Thus, and q(x) = X^TAX, where X = [x,y]^T.

(b) Here xy . Thus, and

(d) Here q(X) = X^TAX and X = PY. Thus, X^T = Y^TP^T. Therefore,

[As expected, the results in parts (c) and (d) are equal.]

12.13. Consider any diagonal matrix A = diag(a₁, … ; a_n) over K. Show that for any nonzero scalars k₁, … ; k_n ∈ K, A is congruent to a diagonal matrix D with diagonal entries a₁k²₁; … ; a_nk²_n. Furthermore, show that

(a) If K = C, then we can choose D so that its diagonal entries are only 1’s and 0’s.

(b) If K = R, then we can choose D so that its diagonal entries are only 1’s, 1’s, and 0’s.

Let P = diag(k₁, … ; k_n). Then, as required,

(a) Let P = diag(b_i), where

Then P^TAP has the required form.

(b) Let P = diag(b_i), where

Then P^TAP has the required form.

Remark: We emphasize that (b) is no longer true if “congruence” is replaced by “Hermitian congruence.”

12.14. Prove Theorem 12.5: Let f be a symmetric bilinear form on V over R. Then there exists a basis of V in which f is represented by a diagonal matrix. Every other diagonal matrix representation of f has the same number p of positive entries and the same number n of negative entries.

By Theorem 12.4, there is a basis {u₁, … , u_n} of V in which f is represented by a diagonal matrix with, say, p positive and n negative entries. Now suppose {w₁, … , w_n} is another basis of V, in which f is represented by a diagonal matrix with p^′ positive and n^′ negative entries. We can assume without loss of generality that the positive entries in each matrix appear first. Because rank(f) = p + n = p^′ + n⁰, it suffices to prove that p = p^′.

Let U be the linear span of u₁, … , u_p and let W be the linear span of w_p0+1, … , w_n. Then f(v, v) > 0 for every nonzero ν ∈ U, and f(v, v) 0 for every nonzero v ∈ W. Hence, U ∩ W = {0}. Note that dim U = p and dim W = n – p⁰. Thus,

But dim(U + W) ≥ dim V = n; hence, p – p⁰ + n – n or p ≥ p⁰. Similarly, p⁰ p and therefore p = p⁰, as required.

Remark: The above theorem and proof depend only on the concept of positivity. Thus, the theorem is true for any subfield K of the real field R such as the rational field Q.

Positive Definite Real Quadratic Forms

12.15. Prove that the following definitions of a positive definite quadratic form q are equivalent:

(a) The diagonal entries are all positive in any diagonal representation of q.

(b) q(Y) > 0, for any nonzero vector Y in Rⁿ.

Suppose . If all the coefficients are positive, then clearly q(Y) > 0 whenever Y ≠ 0. Thus, (a) implies (b). Conversely, suppose (a) is not true; that is, suppose some diagonal entry a_k ≥ 0. Let e_k = (0; … ; 1; … 0) be the vector whose entries are all 0 except 1 in the kth position. Then q(e_k) = a_k is not positive, and so (b) is not true. That is, (b) implies (a). Accordingly, (a) and (b) are equivalent.

12.16. Determine whether each of the following quadratic forms q is positive definite:

(a) q(x; y; z) = x² + 2y² 4xz 4yz + 7z²

(b) q(x; y; z) = x² + y² + 2xz + 4yz + 3z²

Diagonalize (under congruence) the symmetric matrix A corresponding to q.

(a) Apply the operations “Replace R₃ by 2R₁ + R₃” and “Replace C₃ by 2C₁ + C₃,” and then “Replace R₃ by R₂ + R₃” and “Replace C₃ by C₂ + C₃.” These yield

The diagonal representation of q only contains positive entries, 1, 2, 1, on the diagonal. Thus, q is positive definite.

(b) We have

There is a negative entry –2 on the diagonal representation of q. Thus, q is not positive definite.

12.17. Show that q(x, y) = ax² + bxy + cy² is positive definite if and only if a > 0 and the discriminant D = b² – 4ac < 0.

Suppose v = (x,y) 0. Then either x ≠ 0 or y ≠ 0; say, y ≠ 0. Let t ≠ x/y. Then

However, the following are equivalent:

(i) s = at² + bt + c is positive for every value of t.

(ii) s = at² + bt + c lies above the t-axis.

(iii) a > 0 and D = b² – 4ac < 0.

Thus, q is positive definite if and only if a > 0 and D < 0. [Remark: D < 0 is the same as det(A) > 0, where A is the symmetric matrix corresponding to q.]

12.18. Determine whether or not each of the following quadratic forms q is positive definite:

(a) q(x, y) = x² 4xy + 7y², (b) q(x, y) = x² + 8xy + 5y², (c) q(x; y) = 3x² + 2xy + y² Compute the discriminant D = b² 4ac, and then use Problem 12.17.

(a) D = 16 – 28 = 12. Because a = 1 > 0 and D < 0; q is positive definite.

(b) D = 64 – 20 = 44. Because D > 0; – q is not positive definite.

Hermitian Forms

12.19. Determine whether the following matrices are Hermitian:

A complex matrix A = [a_ij] is Hermitian if A* = A—that is, if

(a) Yes, because it is equal to its conjugate transpose.

(b) No, even though it is symmetric.

12.20. Let A be a Hermitian matrix. Show that f is a Hermitian form on Cⁿ where f is defined by Images

For all a, b ∈ C and all X₁, X₂, Y ∈ Cⁿ,

Hence, f is linear in the first variable. Also,

Hence, f is a Hermitian form on Cⁿ.

Remark: We use the fact that is a scalar and so it is equal to its transpose.

12.21. Let f be a Hermitian form on V. Let H be the matrix of f in a basis S = {u_i} of V. Prove the following:

(a)

(b) If P is the change-of-basis matrix from S to a new basis S⁰ of V, then (or B = Q *HQ, where Q = P) is the matrix of f in the new basis S⁰.

Note that (b) is the complex analog of Theorem 12.2.

(a) Let u, v ∈ V and suppose u = a₁u₁ + … + a_nu_n and v = b₁u₁ + + b_nu_n. Then, as required,

Images

(b) Because P is the change-of-basis matrix from S to S⁰, we have P[u]_S′ = [u] _S and P[v] _S0 = [v] _S, hence, Images and Images : Thus, by (a),

But u and v are arbitrary elements of V; hence, . is the matrix of f in the basis S⁰:

12.22. Let Images , a Hermitian matrix.

Find a nonsingular matrix P such that is diagonal. Also, find the signature of H.

Use the modified Algorithm 12.1 that applies the same row operations but the corresponding conjugate column operations. Thus, first form the block matrix M = [H, I]:

Apply the row operations “Replace R₂ by (–1 + i)R₁ + R₂” and “Replace R₃ by 2iR₁ + R₃” and then the corresponding conjugate column operations “Replace C₂ by (–1 –i)C₁ + C₂” and “Replace C₃ by 2iC₁ + C₃” to obtain

Next apply the row operation “Replace R₃ by 5iR₂ + 2R₃” and the corresponding conjugate column operation “Replace C₃ by 5iC₂ + 2C₃” to obtain

Now H has been diagonalized, and the transpose of the right half of M is P. Thus, set

Note D has p = 2 positive elements and n = 1 negative elements. Thus, the signature of H is sig(H) = 2 – 1 = 1.

Miscellaneous Problems

12.23. Prove Theorem 12.3: Let f be an alternating form on V. Then there exists a basis of V in which f is represented by a block diagonal matrix M with blocks of the form Images or 0. The number of nonzero blocks is uniquely determined by f [because it is equal to Images rank(f).

If f = 0, then the theorem is obviously true. Also, if dim V = 1, then f(k₁u; k₂u) = k₁k₂f (u, u) = 0 and so f = 0. Accordingly, we can assume that dim V > 1 and f ≠ 0.

Because f ≠ 0, there exist (nonzero) u₁, u₂ ∈ V such that f(u₁, u₂) ≠ 0. In fact, multiplying u₁ by an appropriate factor, we can assume that f (u₁; u₂) = 1 and so f(u₂, u₁) = 1. Now u₁ and u₂ are linearly independent; because if, say, u₂ = ku₁, then f(u₁, u₂) = f(u₁, ku₁) = kf(u₁, u₁) = 0. Let U = span(u₁, u₂); then,

(i) The matrix representation of the restriction of f to U in the basis Images ,

(ii) If u ∈ U, say u = au₁ + bu₂, then

Let W consists of those vectors w ∈ V such that f(w, u₁) = 0 and f(w; u₂) = 0: Equivalently,

We claim that V = U ⊕ W. It is clear that U ∩ W = {0}, and so it remains to show that V = U + W. Let v ∈ V. Set

Because u is a linear combination of u₁ and u₂, u ∈ U.

We show next that w ∈ W. By (1) and (ii), f(u, u₁) = f (v; u₁); hence,

Similarly, f(u; u₂) = f(u, u₂)

Then w ∈ W and so, by (1), v = u + w, where u ∈ W. This shows that V = U + W; therefore, V = U ⊕ W.

Now the restriction of f to W is an alternating bilinear form on W. By induction, there exists a basis u₃, … , u_n of W in which the matrix representing f restricted to W has the desired form. Accordingly, u₁, u₂, u₃, … ; u_n is a basis of V in which the matrix representing f has the desired form.

SUPPLEMENTARY PROBLEMS

Bilinear Forms

12.24. Let u = (x₁, x₂) and v = (y₁, y₂). Determine which of the following are bilinear forms on R²:

12.25. Let f be the bilinear form on R² defined by

(a) Find the matrix A of f in the basis {u₁ = (1; 1); u₂ = (1; 2)}.

(b) Find the matrix B of f in the basis {v₁ = (1; 1); v₂ = (3; 1)}.

12.26. Let V be the vector space of two-square matrices over R. Let Images , and let f(A, B) = tr(A^TMB), where A, B ∈ V and “tr” denotes trace. (a) Show that f is a bilinear form on V. (b) Find the matrix of f in the basis

12.27. Let B(V) be the set of bilinear forms on V over K. Prove the following:

(a) If f, g ∈ B(V), then f + g, kg ∈ B(V) for any k ∈ K.

(b) If ϕ and σ are linear functions on V, then f(u, v) = f(u)s(v) belongs to B(V).

12.28. Let [f] denote the matrix representation of a bilinear form f on V relative to a basis {u_i}. Show that the mapping f ℯ [f] is an isomorphism of B(V) onto the vector space V of n-square matrices.

12.29. Let f be a bilinear form on V For any subset S of V, let

Show that: (a) S^T and S^T are subspaces of V; (b) Sj C S₂ implies (c) ;

12.30.. Suppose f is a bilinear form on V Prove that: rank( f) = dim V — dim V? = dim V — dim V^T, and hence, dim V? = dim V^T.

12.31. Let f be a bilinear form on V For each u 2 V, let U:V → K and w:V → K be defined by U(x) = f (x, u) and ũ(x) =f (u, x). Prove the following:

(a) ũ and u are each linear; i.e., w, u 2 V*,

(b) u → w and un u are each linear mappings from V into V *,

12.32. Show that congruence of matrices (denoted by ') is an equivalence relation; that is,

(i) A ' A; (ii) If A ' B, then B ' A; (iii) If A ' B and B ' C, then A ' C.

Symmetric Bilinear Forms, Quadratic Forms

12.33. Find the symmetric matrix A belonging to each of the following quadratic forms:

12.34. For each of the following symmetric matrices A, find a nonsingular matrix P such that D = P^TAP is diagonal:

12.35. Letq(x, y) 2X²- 6xy — 3y² and x = s + 2t, y =, 3s - t.

(a) Rewrite q(x, y) in matrix notation, and find the matrix A representing the quadratic form.

(b) Rewrite the linear substitution using matrix notation, and find the matrix P corresponding to the substitution.

12.36. For each of the following quadratic forms q(x, y, z), find a nonsingular linear substitution expressing the variables x, y, z in terms of variables r, s, t such that q(r, s, t) is diagonal:

(a) q(x, y, z) = x² + 6xy + 8y² — 4xz + 2yz — 9z²,

(b) q(x, y, z) = 2x² — 3y² + 8xz + 12yz + 25z²,

In each case, find the rank and signature.

12.37. Give an example of a quadratic form q(x,y) such that q(u) = 0 and q(v) = 0 but q^(u + ^v) = °.

12.38. Let S(V) denote all symmetric bilinear forms on V Show that

(a) S(V) is a subspace of B(V); (b) If dim V = n, then dim .

12.39. Consider a real quadratic polynomial Images where a_ij = a_ji.

(a) If a₁₁ ≠ 0, show that the substitution

yields the equation , where q⁰ is also a quadratic polynomial.

(b) If a₁₁ = 0 but, say, a₁₂ ≠ 0, show that the substitution x₁ = y₁ + y₂;

yields the equation q(x₁, … ; x_n) = P b_ij y_i y_j, where b₁₁ ≠ 0, which reduces this case to case (a).

Remark: This method of diagonalizing q is known as completing the square.

Positive Definite Quadratic Forms

12.40. Determine whether or not each of the following quadratic forms is positive definite:

12.41. Find those values of k such that the given quadratic form is positive definite:

12.42. Suppose A is a real symmetric positive definite matrix. Show that A = P^TP for some nonsingular matrix P.

Hermitian Forms

12.43. Modify Algorithm 12.1 so that, for a given Hermitian matrix H, it finds a nonsingular matrix P for which Images is diagonal.

12.44. For each Hermitian matrix H, find a nonsingular matrix P such that D = P^TH P is diagonal:

Find the rank and signature in each case.

12.45. Let A be a complex nonsingular matrix. Show that H = A*A is Hermitian and positive definite.

12.46. We say that B is Hermitian congruent to A if there exists a nonsingular matrix P such that Images or, equivalently, if there exists a nonsingular matrix Q such that B = Q*AQ. Show that Hermitian congruence is an equivalence relation. (Note: If Images then Images .)

12.47. Prove Theorem 12.7: Let f be a Hermitian form on V. Then there is a basis S of V in which f is represented by a diagonal matrix, and every such diagonal representation has the same number p of positive entries and the same number n of negative entries.

Miscellaneous Problems

12.48. Let e denote an elementary row operation, and let f* denote the corresponding conjugate column operation (where each scalar k in e is replaced by Images in f*). Show that the elementary matrix corresponding to f* is the conjugate transpose of the elementary matrix corresponding to e.

12.49. Let V and W be vector spaces over K. A mapping f : V W → K is called a bilinear form on V and W if

for every a, b ∈ K, v_i ∈ V; w_j ∈ W. Prove the following:

(a) The set B(V; W) of bilinear forms on V and W is a subspace of the vector space of functions from V × W into K.

(b) If {ϕ₁, … ; ϕ_m} is a basis of V* and {σ₁, … ; σ_n} is a basis of W*, then {f_ij : i = 1; … ; m; j = 1; … ; ng is a basis of B(V; W), where f_ij is defined by f_ij(v; w)= f_i(v)s_j(w). Thus, dim B(V; W) = dim V dim W. [Note that if V = W, then we obtain the space B(V) investigated in this chapter.]

12.50. Let V be a vector space over K. A mapping Images is called a multilinear (or m-linear) form on V if f is linear in each variable; that is, for i = 1; … ; m,

where denotes the ith element, and other elements are held fixed. An m-linear form f is said to be alternating if f (v₁, … v_m) = 0 whenever v_i = v_j for i ≠ j. Prove the following: