# ADVANCED LINEAR ALGEBRA PDF

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The primary text for this course is “Linear Algebra and its Applications”, second edition, by Peter D. Lax (hereinafter referred to as [Lax]). The. International Standard Book Number (eBook - PDF). This book to linear algebra involving abstract vector spaces, subspaces, linear. examples, which are usually presented in introductory linear algebra texts presented in advanced linear algebra books, we use “row reduction” proofs.

V as follows. Let a be any scalar. Hence T is invertible. We note that. We now show that invertibility is a generic property for elements in L U. As an illustration. The availability of a norm of L U allows one to perform analysis on L U so that a deeper understanding of L U may be achieved.

W and it follows from 2. In the rest of the section. Hence 2 follows as well. This proves 1. For a positive integer m. We next consider 2. As in calculus. The matrix version of the discussion here can also be easily formulated analogously and is omitted. Show that N is not an open subset of L U.

Here c2 may be interpreted as the length of the vector in the resolution that is taken to be perpendicular to u. Then we study some applications of determinants. We consider the area of the parallelogram formed from using these two vectors as adjacent edges. First we discuss some motivational examples. Next we present the definition and basic properties of determinants.

We now consider volume. To avoid the trivial situation. Then 3. We shall apply some vector algebra over R3 to facilitate our discussion. From 3. Multiplying the first equation by b1. If c is a regular value of f. It is clear that. As a simple application. The advantage of using the integral representation for N f.

We next extend our discussion of topological invariants to two-dimensional situations. In analogy with the case of a real-valued function over an interval discussed above. Let v be a vector field over R2. So for any closed curve. With 3. By continuity or topological invariance. The meaning of this result will be seen in the R following theorem. Theorem 3. Using this property and 3. We now use Theorem 3. Consider the vector field v x. In order to simplify our calculation. Q are real-valued functions of x.

In view of 3. Proof Without loss of generality and for sake of simplicity. Since the orientation of S 2 is given by its unit outnormal vector. To further facilitate computation. For such a choice of R. At the image point u.

We may also consider a map u: Do the results depend on a. Exercises 3. Regarded as the identity map u: There are many ways to. Use the topological method in this section to prove that the system has at least one solution. The above definition of a determinant immediately leads to the following important properties.. The inductive definition to be presented below is perhaps the simplest.

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To show this. Definition 3. Proof We prove the theorem by induction. As an application. Proof We again use induction on n. For A. Proof We use induction on n. Let A. For the adjacent row interchange. Thus adding a multiple of a row to another row of A does not alter the determinant ofA: Let E be an elementary matrix of a given type. In all cases. The matrices constructed in the above three types of permissible row operations are called elementary matrices of types 1.

Ek are some elementary matrices. Such an operation may also be realized by multiplying A from the left by the matrix obtained from adding the same multiple of the row to another row. We now prove that the conclusion of Theorem 3. Proof Using induction. If the column interchange does not involve the first column. Let E1. The effect of the value of determinant with respect to such an alteration needs to be examined closely.

This property is not so obvious since our definition of determinant is based on the cofactor expansion by the first column vector and an interchange of the first column with another column alters the first column of the matrix. Now consider the effect of an interchange of the first and second columns of A. We still use induction. First we show that a determinant is invariant under matrix transpose. The matrices constructed in the above three types of permissible column operations are simply the elementary matrices defined and described earlier.

Like those for permissible row operations. Such an operation may also be realized by multiplying A from the right by the matrix obtained from adding the same multiple of the column to another column. The proof is complete. On the other hand 3. Next we show that a determinant preserves matrix multiplication. There are two cases to be treated separately. Ek be a sequence of elementary matrices so that 3.

Thus 3. As another example.

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Proof We first consider the column expansion case. Below we show that we can make a cofactor expansion along any column or row to evaluate a determinant. In the next section. Now apply the same sequences of permissible row operations on A.

## MATH 4063 Advanced Linear Algebra

The validity of the cofactor expansion along an arbitrary row may be proved by induction as done for the column case. Applying Theorem 3. Establish the formulas to express the determinants of these matrices in terms of the anti-diagonal entries an1. This result is also known as the Minkowski theorem. This result is also known as the Levy— Desplanques theorem.. We can summarize the properties stated in 3. For k. What happens when n is even?

These results motivate the following definition. As a consequence of this definition and 3. The adjugate matrix of A. In such a situation we can use 3. As an important application. To end this section. Use B to denote the submatrix of A consisting of those k row vectors.

To ease the illustration to follow.. Then there are k row vectors of A which are linearly independent. Then C is a submatrix of A which lies in F k. Use C to denote the submatrix of B consisting of those k column vectors. The unique determination of a polynomial by interpolation. Since B is of rank k we know that there are k column vectors of B which are linearly independent. The determinant 3. In view of this fact and the fact that the matrix representations of a linear mapping from a finite-dimensional vector space into itself with respect to different bases are similar.

Under such a condition the coefficients a0. We first consider the linear mapping TA: The eigenvalues. So the eigenvalues of A are the characteristic roots of A. A can have at most n distinct eigenvalues. The purpose of this section is to show how to use determinant as a tool to find the eigenvalues of A. We now consider the abstract case.

Next we consider the general situation when A may have multiple eigenvalues. We begin by showing that A may be approximated by matrices of n distinct eigenvalues.

In view of Theorem 3. We show this by induction on n. First we assume that A has n distinct eigenvalues. Let u1 be an associated eigenvector. Then the linear mapping defined by 3. We have seen that the proof of Theorem 3. Of course Theorem 3. By the inductive assumption. In the rest of this section we shall assume that the variable of any polynomial we consider is such a formal symbol. It is clear that this assumption does not affect the computation we perform.

The set of all polynomials with coefficients in F and in the variable t is denoted by P. Given a field F. Therefore the Cayley— Hamilton theorem may be restated in terms of linear mappings as follows. What is the characteristic polynomial of T? Definition 4. A scalar product over U is defined to be a bilinear symmetric function f: We say that u. We shall start from the most general situation of scalar products.

Furthermore it is easy to show that if the vectors u1. In other words the following properties hold. We then consider the situations when scalar products are non-degenerate and positive definite. Then we can resolve v into the sum of two mutually perpendicular vectors. Theorem 4. So we have seen that we may assume v1. Any basis of U0 can be extended to become an orthogonal basis of U.

If wi. In other words any two vectors in such a basis of U are mutually perpendicular. Without loss of generality.

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The method described in the proof of Theorem 4. We may assume w3. Such a statement is also known as the Sylvester theorem. It is clear that for the orthogonal basis 4. Show that if these vectors are not null then they must be linearly independent. These integers are sometimes referred to as the indices of nullity.

So the asserted linear independence follows. Inserting this result into 4. Thus the integers n0. Exercises 4. W two subspaces of U. Since ui. In order to make our discussion more precise. Such a statement. Thus T is orthogonal. We say that T is an orthogonal mapping if T u.

We express any u. That T being orthogonal implies 2 is trivial. T un are mutually orthogonal and T ui. Proof If T is orthogonal. As an immediate consequence of the above definition. Now assume 1 is valid. Assume 2 holds. Using the properties of the scalar product.

So 3 follows. That 3 implies the orthogonality of T is obvious. In view of the discussion in Section 2. This example clearly illustrates the dependence of the form of an orthogonal mapping on the underlying scalar product. We now show that T being orthogonal and 3 are equivalent. Decompose U into the direct sum as stated in 4. Real ones are modeled over the standard Euclidean scalar product on Rn: Use 4.

W be two subspaces of U. Positivity u. Motivated from the above examples. Partial homogeneity u. Needless to say. The major difference is that the former is symmetric but the latter fails to be so. A positive definite scalar product over a complex vector space U is a scalar function u. Since the real case is contained as a special situation of the complex case.

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Thus 4. The inequality 4. Let u. That it also satisfies the triangle inequality will be established shortly. Then in view of 4. For u. Of course 4. Such an orthogonal basis is called an orthonormal basis. Since w depends on v. V be finite-dimensional vector spaces over C with positive definite scalar products. Hence v. Of particular interest is a mapping from U into itself.

Similar to Definition 4. Assume 1 holds. In analogue to Theorem 4. The rest of the proof is similar to that of Theorem 4. The above discussion may be carried over to the abstract setting as follows.

If TA is orthogonal. If TA is unitary. It is easily checked that A is orthogonal if and only if its sets of column and row vectors both form orthonormal bases of Rn with the standard Euclidean scalar product. The above calculations lead us to formulate the following concepts. With the above terminology and the Gram—Schmidt procedure. Then u.

Similar to the real vector space situation. If T is orthogonal. If T is self-adjoint. It is easily checked that A is unitary if and only if its sets of column and row vectors both form orthonormal bases of Cn with the standard Hermitian scalar product.

In this case. From the Schwarz inequality 4. Show that u1. Show that A. It is clear that if A is real then the matrices Q and R are also real and Q is orthogonal. Establish the following statement known as the Fredholm alternative for complex matrix equations: With such a scalar product. We begin with the following basic orthogonal decomposition theorem. We focus our attention on the problem of resolving a vector into the span of a given set of orthogonal vectors.

We now follow 4. From u. By the positivity condition. For the scalars a1. The set is said to be complete if it is a basis of U. Of particular interest is when a set of orthogonal vectors becomes a basis. So the proof is complete. The completeness of a set of orthogonal vectors is seen to be characterized by the norms of vectors in relation to their Fourier coefficients.

Proof For given u. It is interesting to note that. Use PV: We show that T must be linear. A mapping T from U into itself satisfying the isometric property 4. A mapping from U into itself satisfying such a property is called an isometry or isometric.

So homogeneity is established and the proof follows. The vector space we consider here is taken to be Cn with the standard Hermitian scalar product 4. Replacing u by u where m is a nonzero integer. We next give an example showing that.

Using this result. Then T satisfies 4. It will be interesting to spell out some conditions in the complex situation in addition to 4.

The following theorem is such a result. We now need to show that the converse is true. In view of the additivity property of T. Therefore 4. Establish the following variant of Theorem 4.

In view of these and 4. It is worth noting that Theorems 4. A mapping T from U into itself satisfying the property 4. After these we study the commutativity of self-adjoint mappings. In the last section we show the effectiveness of using self-adjoint mappings in computing the norm of a mapping between different spaces and in the formalism of least squares approximations.

We then establish the main spectrum theorem for self-adjoint mappings based on a proof of the existence of an eigenvalue using calculus. Definition 5. The next simplest real-valued functions to be studied are bilinear forms whose definition is given as follows. We first present a general discussion on bilinear and quadratic forms and their matrix representations. The simplest real-valued functions over U are linear functions. We next focus on characterizing the positive definiteness of self-adjoint mappings.

We also show how a symmetric bilinear form may be uniquely represented by a self-adjoint mapping. For a bilinear form f: Section 1. Of course q is uniquely determined by f through 5. To proceed. Let f: This observation motivates the following definition.

From now on we will concentrate on symmetric bilinear forms. A defines a self-adjoint mapping over Rn with respect to the standard Euclidean scalar product over Rn. In a similar manner. Recall that x. In a more precise manner. Since f is bilinear. Note that. B must have the same rank. B are congruent then A. The self-adjointness or symmetry of T follows from the symmetry of f and the scalar product. B are congruent and A is symmetric then so is B. Show that if A. For any symmetric bilinear form f: Theorem 5.

Exercises 5. Then 1 T must have a real eigenvalue. Proof We use induction on dim U. For self-adjoint mappings. Thus we have we see that x Combining 5. Then we have. Using the inductive assumption. Using Theorem 5. Hence T u. A useful matrix version of Theorem 5.

This observation suggests a practical way to construct an orthogonal basis of U consisting of eigenvectors of T: First find all eigenspaces of T. Hence there are column vectors u1. It is clear that n0 is simply the nullity of T. Finally put all these orthogonal bases together to get an orthogonal basis of the full space. Note that the proof of Theorem 5.

Then obtain an orthogonal basis for each of these eigenspaces by using the Gram—Schmidt procedure. Consider A as an element in C n. Proof The right-hand side of 5. Prove that A is orthogonal. If A is orthogonal. Therefore it will be sufficient to study the positive definiteness of self-adjoint mappings which will be shown to be equivalent to the positive definiteness of any matrix representations of the associated bilinear forms of the self-adjoint mappings.

Proof Assume T is positive definite. In view of 1. Suppose that A is positive definite. So S is invertible. So the equivalence of the positive definiteness of T and the statement 4 follows. Therefore u. Now assume 5. We now show that the positive definiteness of T is equivalent to the statement 3. So 5 follows. It is obvious that 2 implies the positive definiteness of T.

Proof That A is positive definite is equivalent to either 1 or 2 has already been demonstrated in the proof of Theorem 5. For practical convenience. Hence 5 implies 4 as well.

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Thus 4 holds. So 6 follows. A is positive semi-definite but not positive definite. For positive semidefinite or non-negative mappings and matrices. Define the diagonal matrix D1 by 5.

A is indefinite. Prove that if A is of full rank. A is negative semi-definite but not negative definite. At A is positive definite and the other is positive semi-definite but can never be positive definite. Prove that the eigenvalues of AB are all positive.

B are greater than or equal to a. Assume that A. Hence S is the null-space of the matrix A. What happens when A is indefinite such that there are nonzero vectors x. A and B are simultaneously congruent to diagonal matrices. The first one involves using determinants and the second one involves an elegant matrix decomposition. Prove that q is convex. The matrix A is positive definite if and only if all its leading principal minors are positive.

Therefore the proof is complete. We next assume that 5. The matrix A is positive definite if and only if all its principal minors are positive. The quantity det Ai We now pursue the decomposition of a positive definite matrix as another characterization of this kind of matrices. It is clear that if A is positive definite then so is Ai Therefore we arrive at the following slightly strengthened version of Theorem 5. Hence det Ai So it remains to show that the number a may be uniquely determined as well.

We again use induction. In view of Theorem 5. Therefore the last relation in 5. Can you extend your finding to the case over Rn? Find a necessary and sufficient condition on a1. Prove that if the matrix A is positive semi-definite then all its leading principal minors are non-negative. The main focus of this section is to characterize a situation when two mappings may be simultaneously diagonalized. Note that although the theorem says that.

Then S commutes with any mapping and any nonzero vector u is an eigenvector of S but u cannot be an eigenvector of all self-adjoint mappings.

The matrix version of Theorem 5. Then there is an orthonormal basis of U consisting of eigenvectors of both S.

Assume that S. Two symmetric matrices A. Since S is selfadjoint. T if and only if S and T commute. DB are diagonal matrices in R n. An immediate consequence of 5.

We now show how to extend 5.

Then the Schwarz inequality 4. Proof 5. From 5. Hence 5. An important consequence of Theorem 5. It is not hard to see that the equation 5. V in order for Theorem 5. Hence we may expand the right-hand side of 5. Consider the optimization problem. It is interesting that we do not require any additional properties for the vector spaces U. We next study another important problem in applications known as the least squares approximation. We show that x solves 5. V be finite-dimensional vector spaces with positive definite scalar products.

Using knowledge about self-adjoint mappings. Inserting this into 5. Then a solution x of 5. Formulate a solution of the following optimization problem. We begin with a discussion on the complex version of bilinear forms and the Hermitian structures. We also show how to use self-adjoint mappings to study a mapping between two spaces. We explore the commutativity of self-adjoint mappings and apply it to obtain the main spectrum theorem for normal mappings.

Extending the standard Hermitian scalar product over Cn. We next focus again on the positive definiteness of self-adjoint mappings. We will relate the Hermitian structure of a bilinear form with representing it by a unique selfadjoint mapping.

Then we establish the main spectrum theorem for self-adjoint mappings. Definition 6. Then As in the real situation. Define q: This fact is in sharp contrast with the real situation.

For any column vectors x. In view of 6. Thus we are led to the following definition. Then f is Hermitian if and only if T is self-adjoint. Proof If f is Hermitian. Theorem 6. The converse is similar. Then f may also be represented as f u.

Applying 6. Given any sesquilinear form f: Give an example to show that the same may not hold for a real vector space with a positive definite scalar product. Therefore T is self-adjoint if and only if the matrix representation of T with respect to any orthonormal basis is Hermitian. Show that f is Hermitian if and only if A is Hermitian. Of course the converse is true too. Then T may be reduced over the direct sum of mutually perpendicular eigenspaces.

Show that T is self-adjoint or Hermitian if and only if u. Exercises 6. This establishes 1. From 1. To establish 2. Note that this proof does not assume that U is finite dimensional. Proof If 6. To establish 3. With respect to B. Now since the basis transition matrix from B0 into B is unitary.

With the mapping T defined by 6. Using Theorem 6. This is an extended version of Exercise 4. Show that det A must be a real number. Therefore the positive definiteness of a self-adjoint or Hermitian mapping is central. The proof of Theorem 6. Here we check only that the matrix A defined in 6. Parallel to Theorem 5.

To see how this is done. The proof is similar to that of Theorem 5. The sufficiency proof is also similar but needs some adaptation to meet the delicacy with handling complex numbers. Positive semi-definiteness and negative definiteness can be defined and investigated analogously as in the real situation in Section 5. Such a property suggests that it may be possible to extend Theorem 5. Thus the proof is complete. Rewrite A in the blocked form 6. Applying Theorem 6. Assume now A is positive definite.

Thus it remains only to show that the number a may be uniquely determined as well. Thus the third equation in 6. Show also that. It is not hard to see that the conclusion in this extended situation is the same as in the real situation. We now use Theorem 6. Show also that the same conclusion is true for some orthogonal matrices P and Q in R n. The proof of the theorem is the same as that for the real situation and omitted.

Show that there exist unitary matrices P and Q in C n. Then T is normal if and only if there is an orthonormal basis of U consisting of eigenvectors of T. Proof Suppose that U has an orthonormal basis consisting of eigenvectors of T. This discussion leads us to the following theorem. P and iQ are commutative. T is normal. Then there is an orthonormal basis of U consisting of eigenvectors of S and T simultaneously if and only if S and T are commutative.

Since S is normal. B is diagonal. B are commutative: To prove the theorem. Show that T enjoys a similar polar decomposition property such that there are a positive semi-definite element R and a unitary element S. This is an extended version of Exercise 6. Show that A must be diagonal. As in Section 5. So there is a unique element in U depending on v.

We can similarly show how to extend 6. This result may conveniently be summarized as a theorem. Then we have T ui. In view of this and 6. Now set 1 T ui.

Q are some unitary matrices and B. Note that Exercise 6. Of course the results above may also be established similarly for mappings between real vector spaces and for real matrices. C some positive semi-definite Hermitian matrices. Show that the singular values of T are simply the positive eigenvalues of T. Note that most of the exercises in Chapter 5 may be restated in the context of the complex situation of this chapter and are omitted.

Let g1. Definition 7. Finally we prove the Jordan decomposition theorem by understanding how a mapping behaves itself over each of its generalized eigenspaces. As a preparation we first recall some facts regarding factorization of polynomials.

Hence an ideal is also a subspace. Then we show how to reduce a linear mapping over a set of its invariant subspaces determined by a prime factorization of the characteristic polynomial of the mapping. Next we reduce a linear mapping over its generalized eigenspaces. Since F may naturally be viewed as a subset of P. The following theorem establishes that any ideal in P may be generated from a single element in P.

Theorem 7. Such a polynomial. There are two trivial ideals: Such an observation indicates what to look for in our proof. If the coefficients of the highest-degree terms of g and h coincide. We may say that this deal is generated from g1. Then there are elements f1.

The theorem is proved. In view of Theorem 7.

So g itself is a common divisor of g1. This fact will be our starting point in the subsequent development. Two polynomials are said to be equivalent if one is a scalar multiple of the other. Given g1. Show that gcd f. Exercises 7. In Chapter 5 we focus on real vector spaces with positive definite scalar products and quadratic forms.

We first establish the main spectral theorem for self-adjoint mappings. We will not take the traditional path of first using the Fundamental Theorem of Algebra to assert that there is an eigenvalue and then applying the self-adjointness to show that the eigenvalue must be real. Instead we shall formulate an optimization problem and use calculus to prove directly that a self-adjoint mapping must have a real eigenvalue.

We then present a series of characteristic conditions for a symmetric bilinear form, a symmetric matrix, or a self-adjoint mapping, to be positive definite. We end the chapter by a discussion of the commutativity of self-adjoint mappings and the usefulness of self-adjoint mappings for the investigation of linear mappings between different spaces.

In Chapter 6 we study complex vector spaces with Hermitian scalar products and related notions. Much of the theory here is parallel to that of the real space situation with the exception that normal mappings can only be fully understood and appreciated within a complex space formalism. In Chapter 7 we establish the Jordan decomposition theorem. We start with a discussion of some basic facts regarding polynomials. We next show how Preface to reduce a linear mapping over its generalized eigenspaces via the Cayley Hamilton theorem and the prime factorization of the characteristic polynomial of the mapping.

We then prove the Jordan decomposition theorem. The key and often the most difficult step in this construction is a full understanding of how a nilpotent mapping is reduced canonically.

We approach this problem inductively with the degree of a nilpotent mapping and show that it is crucial to tackle a mapping of degree 2. Such a treatment eases the subtlety of the subject considerably. In Chapter 8 we present four selected topics that may be used as materials for some optional extra-curricular study when time and interest permit. In the first section we present the Schur decomposition theorem, which may be viewed as a complement to the Jordan decomposition theorem.

In the second section we give a classification of skewsymmetric bilinear forms. In the third section we state and prove the PerronFrobenius theorem regarding the principal eigenvalues of positive matrices.

In the fourth section we establish some basic properties of the Markov matrices. In Chapter 9 we present yet another selected topic for the purpose of optional extra-curricular study: a short excursion into quantum mechanics using gadgets purely from linear algebra. Specifically we will use Cn as the state space and Hermitian matrices as quantum mechanical observables to formulate the over-simplified quantum mechanical postulates including Bohrs statistical interpretation of quantum mechanics and the Schrdinger equation governing the time evolution of a state.

We next establish Heisenbergs uncertainty principle. Then we prove the equivalence of the Schrdinger description via the Schrdinger equation and the Heisenberg description via the Heisenberg equation of quantum mechanics. Also provided in the book is a rich collection of mostly proof-oriented exercises to supplement and consolidate the main course materials.

The diversity and elasticity of these exercises aim to satisfy the needs and interests of students from a wide variety of backgrounds. At the end of the book, solutions to some selected exercises are presented. These exercises and solutions provide additional illustrative examples, extend main course materials, and render convenience for the reader to master the subjects and methods covered in a broader range.

Finally some bibliographic notes conclude the book. This text may be curtailed to meet the time constraint of a semester-long course. Here is a suggested list of selected sections for such a plan: Sections 1.A can have at most n distinct eigenvalues.

In the third section we state and prove the Perron—Frobenius theorem regarding the principal eigenvalues of positive matrices. We begin by stating the definition and a discussion of the structural properties of linear mappings. It may also be examined that mapping addition and scalar-mapping multiplication correspond to matrix addition and scalar-matrix multiplication. This example clearly illustrates the dependence of the form of an orthogonal mapping on the underlying scalar product.

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