As can be seen above, the first index denotes the row and the second one - the column, so the matrix entry aij fills the cell at the i-th row and j-th column. These entries may be numbers, functions or any kind of retrievable data. Instead of 6. Note that this is a straightforward generalization of the linear laws for vec- tors introduced above.
In this context, it seems reasonable to view matrices as ordered systems of vector-rows or vector-columns, respectively.
Such un- derstanding justifies also the way we define multiplication of two matrices. Among the many possibilities to do that, people have chosen to work with the so-called row-by-column rule. It consist of the following: the first matrix factor A is considered as a system of vector-rows and the second one B, as a system of vector-columns. Then, the common middle index is factored out in the summation process and the dimension of the product is determined by what is left.
Special Types of Matrices The whole algebra of matrices is something vast and it is convenient to consider only certain subalgebras, i. Note that the two operations impose different restrictions on the matrix dimensions, which can be satisfied simultaneously only if the number of rows and columns are equal, i. Clearly, both the sums and products of square matrices of the same dimension are well-defined.
This peculiar type of skew-symmetric prod- uct defines a different algebraic structure over the linear space of matrices, known as Lie algebra, which lies in the foundation of modern theoretical physics. In other words, these are matrices with only zero en- tries respectively below or above the main diagonal, i. The 1 in particular, the existence of AB does not imply the existence of BA and vice versa. They also form a subalgebra, which is commutative: this is actually the algebra of the complex numbers in a matrix representa- tion.
Transposition and Conjugation We may introduce additional operations in the algebra of matrices. Such a generalization is useful for the definition of real scalar product for complex-valued vectors. Can you think of an example in C2? In the case of square matrices, transposition acts as a reflection about the main diagonal. Naturally, there are symmetric matrices, which are preserved by this reflection, i. Besides, these matrices anti- commute, e.
They are associative but non-commutative and thus, constitute a a more general object, which is not a field the largest number field is C. We also have dual quaternions, whose coefficients are dual numbers, or split quaternions, for which some matrices in the above basis become hermitian.
All those objects find many applications in various areas of science and technology: from classical and quantum mechanics to virtual reality and computer vi- sion.
They also present simple examples of Clifford geometric algebras, which have proven to provide a more efficient description of the physical reality compared to the classical approach. Note also that matrices find application in practically any field which deals with data that is any field, which allows for measurement.
For example, we may use them in modeling stochastic processes with conditional proba- bilities: aij denotes the probability that the event i takes place under the condition j.
Matrix multiplication then yields the correct overall proba- bilities for compound processes, e. Similarly, aij may be interpreted as transition coefficients and the entries of the squared matrix - as the corresponding coefficients for two-step transitions, A3 yields the three-step transitions and so on. Then, it is no wonder that matrices are being constantly used in all kinds of simulations and optimization problems.
Calculate all the commutators of the matrices 6. Show that each hermitian matrix may be decomposed into a sum of a real symmetric matrix and a purely imaginary skew-symmetric one.
Show that each strictly upper lower -triangular matrix 3 is nilpotent, i. Can you find the value of the order m? What is the geometric interpretation of its coefficients? Show the transitivity of the vanishing commutator, i. We use this idea below to transform higher-dimensional matrices into triangular form an thus, express their determinants simply as products of the corresponding diagonal entries.
Note that the first indices in the three factors of each term are ordered fixed to 1, 2 and 3 , while the second ones exhaust all the even and odd permutations with positive and negative sign, respectively. The number of summands, however, increases rapidly with n, e.
Higher-Dimensional Determinants One standard way of calculating determinants in arbitrary dimension is based on the property that for triangular matrices the only non-zero term in the corresponding expression is the product of all diagonal elements, i. This certainly holds for columns too, but we are not allowed to manipulate both simultaneously. This technique is used to trans- form numerical matrices into triangular form and thus obtain their deter- minants.
To begin with, we want to eliminate the first components of all ak except a1. On each successive step we preserve one more of the transformed vectors and use it to eliminate the first non-zero component of the remaining ones.
The co-factor Aij is then equal to this minor, up to a sign. First of all, the one using triangular matrices is surely more effi- cient in terms of computational operations and thus, preferred in numerical algorithms. However, it cannot cope with symbolic expressions, as it uses numerical values explicitly. This difference persists in the context of linear matrix equations and we devote the following example to its clarification. Problem Session 1. Calculate the determinants x2.
For the applica- tions it is often important to know if possible how to invert this operation, i. It is natural to ask if all matrices are invertible, i. Actually, that is one of the two main reasons to consider determinants in the first place.
This vanishing expression is called a fake expansion and we may include it in formula 6. The only problem is that summation should be taken over the middle indices as formula 6. We investigate this matter more thoroughly in the following chapter and relate it to the solution of linear matrix equations. Problem Session: 1. This construction is naturally generalized to the n-dimensional case. Nevertheless, the above formula suggests a closed form solution that is a major advantage compared to the purely numerical Gauss method especially in the case of parameter-dependent systems, in which it is often the only tool for obtaining the solution.
As we learned from working with determinants, such transformations allow for obtaining the matrix in a triangular form. This is the so-called Gauss method. In the regular case one may proceed with the elimination from the bottom up, thus transforming the matrix into a diagonal form and then, dividing each row by its diagonal element, obtain the identity matrix on the left.
Then, on the right, one ends up with the solution, i. The whole procedure is usually referred to as the Gauss-Jordan method elimination. Note that unlike in the case of determinants, here we are allowed to swap rows and multiply them by arbitrary non-zero constants.
On the other hand, we cannot work with columns as this alters the equations. Thus, we end up with two equations for three variables, which cannot lead to an unique solution and we introduce a parameter, e. But then, the whole system has no solution since its equations contradict with each other.
Such systems are called inconsistent. We devote the following section to the study of determined, dependent and inconsistent systems. Two such planes are either parallel in particular coincident or intersect in a line. In the regular case the two infinite lines corresponding to the intersection of pairs of equations in the system, themselves intersect at a point, which gives the solution.
These two lines, on the other hand, may also coincide in the dependent case , or be parallel in the inconsistent one. Moreover, two or even all three of the planes may be parallel or coincident, which corresponds to either inconsistency or a two-parameter degenerate solution, respectively.
These settings are intrinsically related to the notion of rank discussed below. In the singular case, on the other hand, i. The following theorem makes use of the notion of rank that has various definitions in literature, some of which sound quite puzzling.
We first define the rank of a system of vectors as the dimension of their linear span that is certainly equal to the maximal number of linearly independent vectors within the system. The rank of a matrix A is then naturally defined as the rank of the system of its vector- rows or vector-columns.
In other words, a system of vectors in Rn can never have a linear span which exceeds the ambient space. The existence of bases in Rn is a straightforward consequence allowing each vector to be expressed as a linear combination of a system with maximal rank. Then, according to the above theorem, the maximal rank it can have is min n, m , i.
Thus, one may always isolate a square sub-matrix, which contains all the information about the rank and square matrices have maximal rank exactly when the determi- nant is non-vanishing, i. The invariance of the definition with respect to matrix transpositions demands that there is exactly one linear relation between the vector-columns too. Indeed, it is not difficult to see that the third column may be obtained as a sum of the first two.
In this case each pair of vector-rows or columns spans a plane all minors of order two are non-vanishing , but the three vectors cannot span a three-dimensional space. Moreover, if the conditions of the theorem are satisfied, the number of unknowns minus the number of independent equations is equal to the number of free parameters in the solution.
For example, in the regular case the two numbers are equal and the solution determines a point in Rn no free parameters. In the dependent case, on the other hand, we may have k independent relations between the vector-rows or columns of A. In the former case, the system is either over-determined, or inconsistent, while in the latter, it might only be dependent or inconsistent.
Fortunately, the Kronecker-Capelli allows to distinguish between those possibilities here as well. As we shall see later, each linear equation for x, y and z has the interpretation of a plane in R3. Then, the solutions to 3 linear systems correspond to intersections common points, if any of three such planes, and the Kronecker-Capelli theorem allows us to distinguish between different cases see Figure 7.
This theorem remains valid in the more general context of matrix equations considered in the following section. One way to think of it is as of a system of systems of linear equations, for which the unknown vectors are given by the columns of X and the righthand sides - by the corresponding columns of the matrix B.
Such interpretation allows for applying directly most of the results obtained above. In particular, if A is invertible, one may transform the augmented matrix in such a way that the identity matrix is on the left. Then, on the right one has the unique solution, i. Linear Homogeneous Equations We consider a particular type of matrix equations or systems of linear equations with a vanishing righthand side, referred to as homogeneous.
In this setting our work is greatly simplified, e. One more convenience provided by the homogeneous case is that we do not need to bother about the extensions when applying the Gauss algorithm as it simply does not affect them. Another property of linear homogeneous systems is given by the following Theorem 9 The solutions of a linear homogeneous matrix equation and in particular, system of linear equations constitute a vector space.
Suppose also that A has maximal rank, i. Note the two identity matrices belong to vector spaces of different dimension, so their equality is possible and unavoidable only in the case of square matrices. Note, however, that the Kronecker-Capelli theorem still applies to this case and the rank of A B may exceed that of A, since we add more columns in the extension.
Therefore, one needs to check first whether the two ranks agree. On the other hand, the familiar Gauss method works here with almost no modifications. Then, the remaining rows will either be trivial, in which case the system is simply over-determined, or contain a contradiction and thus, yield inconsistency.
Let us illustrate this with several examples. Provide an example, in which the inequality does not hold. What is the explicit relation between the two? Chapter 8 Vectors in R3 8. For example, linear combinations are the most fundamental operations in any vector space and the notion of linear dependence remains identical, i.
Moreover, the number of independent vectors in a given system usually re- ferred to as its rank cannot exceed the dimension n of the corresponding space. Now, consider the expansion 8. This crucial difference between upper and lower indices remains hid- den in the case of orthonormal bases, in which the two types of coordinates coincide and the transition matrix T is clearly equal to its contragradient.
Such transformations are called orthogonal and they play a central role in both geometry and mechanics. As it turns out, the positive sign of the determinant corresponds to an orientation-preserving transformation, which in our case reduces to a pure rotation.
The product is obviously skew-symmetric by construction why? Note that in each case the double cross product lies in the plane determined by the two vectors in the brackets explain why and prove the equalities!
There is one more generalization of the wedge product in R2. This time we put the emphasis on the fact that it yields the oriented area determined by its two factors. One consequence is that coplanar vectors have vanishing triple product. However, we are going to need normalization factors here too, and it is not difficult to see that the triple product, i.
Note also that the concept of duality is always mutual, i. The dual basis technique naturally extends to higher dimensions as well, but then one does not have the convenience of using the cross product.
Try to formulate an analogous construction for Rn based on the exterior product. Prove the particular case of formula 8. Find also the coordinates of the geometric center, the total surface area and the distance from the point D to the plane determined by A, B and C.
Did you have fun? How about a two-parameter set? Try to explain this coincidence algebraically. However, one should be careful, since there are differences too. To begin with, we may still consider two types of vectors - attached and free, typically represented by radius-vectors of points and their differences. The latter constitute the vector space R3 , while the former, the affine space R3 as a space of points.
Thus, one may think that there are two copies of R3 involved in our geometric considerations. When lines and planes come into play, more complicated spaces appear naturally in the model, such as Grassmannian and projective spaces, but we are not going to discuss them here explicitly.
For example, the vector parametric equation of a line in R3 is written in the exact same way as in R2 See Figure 9. Note, however, that here, unlike in the two-dimensional case we have two, rather than one linear equation.
The above is actually referred to as the general equation of a plane and its geometric interpretation is quite similar to that of a general equation of a line in R2 See Figure 9. Similarly, if we generalize the intercept-intercept equation of a line 5. Things remain relatively simple as long as there are only points and vectors to consider.
Then, we also have incidence relations, e. As for the parametric case 9. Let us now consider the mutual position of two lines: g, determined by its direction vector t and a point A, and h - by the pair s and B, respectively. Similarly, two non-parallel planes intersect at a line, which is the one-parameter solu- tion to the system of two equations for three variable. In particular, this distance may be zero if the two planes coincide or the line belongs to the plane, respectively.
Finally, let us consider the case of three planes in R3 from the perspective of systems of linear equations and the Kronecker-Capelli theorem in partic- ular. In the non-degenerate case, in which the system of the three normals has maximal rank, the planes are bound to intersect at a single point much like the coordinate planes OXY , OY Z and OZX intersect at the origin. The other extreme setting involves parallelism of two or even three normals corresponding to rank two and rank one, respectively.
In between, how- ever, we see something with no analogue in R2 : neither two of the normals are parallel and yet, the system is either dependant of inconsistent, i. Consider other possible settings. Now, the rank of the system the above vectors constitute determines whether the four points are coplanar or in a generic position. Another typical task in stereometry is to determine the mutual position and distance between two lines. Let us take for example the one passing through the pair of points A and B we shall denote it with g and another one say, h containing C and D.
See if it yields the same result. For a better understanding of the geometric properties of operators it is convenient to introduce the notions of range and kernel.
Obviously, due to linearity, ker A always contains the zero in V and is said to be trivial if it does not contain anything else. In this case A is called 1 projecting spacial vectors onto the XY -plane we lose track of the vertical component. A non-trivial example would be a reflection or a rotation in the plane - it covers all R2 and the image of each non-zero vectors is also a non-zero vector.
More generally, as we shall see later, each square matrix with non-vanishing determinant may be thought of as a matrix of a linear isomorphism.
Proof: We only give a sketch of the proof here. In the next section we provide more explicit geometric examples. Since this course is focused mainly on Euclidean geometry, we shall pay special atten- tion to operators related to planar and spacial motions.
In order to do this we only need to note that the parallel to u is preserved by Mu , while the one in the normal direction changes its sign, i.
On the other hand, spacial reflections are usually taken with respect to a plane, or some more general mirror surface. Actually, our construction can be used directly only for free vectors, not for radius-vectors explain why and we already know a straightforward way to obtain them in terms of coordinate differences.
There is another, rather physical, setting that involves reflection with respect to a vector, rather than a plane. In order to obtain the trajectory of a particular reflected ray or elastically scattered particle, e. In physical considerations one may introduce a time parameter t and divide There are more examples of this type at the end of the present chapter. Rotations and the Galilean Group As we already mentioned, all spacial motions are generated by reflections.
Since translations are reduced to vector summation, i. These properties turn the set of planar rotations into a commutative matrix group, i. Adding translations to that, which is allowing each radius-vector to be shifted arbitrarily, one obtains the group of Euclidean motions in the plane.
Note, however, that the translational and rotational components of such motions are intertwined and hence, do not commute. The block-matrix technique works similarly in every dimension and the exten- sion of the translational counterpart is trivial, so the main difference is the group of spacial rotations compared to the planar case.
To begin with, it is non-commutative, which makes it far more difficult to study. Each spacial rotation is again rotation in a plane or about the axis, perpendicular to that plane , which cannot be said for higher dimensions. This plane, however, is not fixed but varies within the group. In particular, rotations about the coordinate axes adopt a simple form, e. Baring in mind the above considerations about the rotation group in R3 , one may still express spacial motions in pretty much the same block-matrix form as in R2 , i.
Note that even if the rotation group is commutative, which is only in R2 , introducing translations to it certainly destroys this property as the above composition formula clearly indicates. This is because translations and ro- tations are entangled in the Galilean motion group in a structure referred 3 clearly, the cross product is responsible for the non-commutativity and when the two axes are collinear the expression is reduced to an addition formula for the tangent function.
This peculiar behavior may be encoded in a conveniently chosen algebraic struc- ture, which naturally adopts similar properties. In mechanics this is also known as screw calculus. Similar algebraic con- structions often make the description of cumbersome geometric and phys- ical theories way easier and more productive, e. Complex numbers, quaternions and groups themselves provide a good illustration of this idea.
Determine the rank and the nullity of a linear operator that projects n-dimensional vectors onto a k-dimensional subspace of Rn. Given the four points A 1; 2; 3 , B 2; 3; 1 , C 3; 1; 2 and D 0; 0; 0 , find the coordinates of the reflections of the geometric center of the pyramid ABCD with respect to its four sides. Find the trajectory of the ball after the reflection. This is especially useful in me- chanical considerations including optimization, determining states of equi- libria for complex systems etc.
However, we are interested only in non- trivial solutions, which we refer to as eigenvectors. In particular, the construction of of eigenvectors and eigenval- ues turns out to be of crucial importance for the study of skew- symmetric, orthogonal and unitary operators with their geometric and dynamical prop- erties. For the description of the nontrivial solutions to equation With the aid of condition For a simple root the corresponding solution contains one undetermined pa- rameter, which corresponds to the freedom of scaling v.
We shall clarify this part by the end of the chapter. We begin with a simple but powerful Theorem 12 The eigenvalues of symmetric and hermitian operators are all real and the eigenvectors corresponding to different eigenvalues - orthogonal. If A is hermitian, we also have from the conjugated version of formula The symmetric case follows as a subset. In the above cases the eigenvectors constitute an orthonormal basis1 , in which the matrix of the corresponding linear operator has a diagonal form.
The orthogonal case obviously follows as a subset. Let f be a smooth function. Therefore, the matrices A and f A commute, so they determine identical sets of eigenvectors prove this point alone! Furthermore, for non-smooth functions f , one may always find an approxi- mations with smooth ones and all properties we need endure the limit.
We shall leave the details of the proof to the reader providing only an motivational example. Now, if two matrices are diagonal in the same basis, they obviously commute and vice versa - if one of them is diagonal in a given basis and the commutator vanishes, the other needs to be diagonal as well if this statement is not obvious to you, work out the details.
Our main example is the exponential function, which maps skew-symmetric skew-hermitian matrices into orthogonal unitary ones. This is also one more way to see that the eigenvalues of the latter lie on the unit circle since for the former they are purely imaginary. Moreover, we may prove that each rotation, i. No axis in R4 is preserved by its action, only the origin the rotation center remains fixed.
In every other case we may encounter rotation in a plane at best, but this is something that does not happen often as it demands a set of highly non-trivial conditions to be satisfied by the corresponding matrix entries. Similarly, we have invariant planes and higher-dimensional subspaces in Rn. How- ever, since the whole plane spanned by v1 and v2 which, by the way, is the orthogonal complement of v3 is preserved by the action of A, we may choose an arbitrary pair of vectors in this plane and in particular, an orthonormal one.
Similarly, if A is a matrix of a bilinear form, i. If we switch from one orthonormal basis to another, as we do in our example, the matrix T is orthogonal and the two transformations would be identical, but this is not so in the generic case. Of course, not.
In this case, the matrix we are dealing with is neither symmetric, nor skew- symmetric and its eigenvectors do not necessarily constitute a basis in R3.
Actually, it is not difficult to see that its rank is equal to two: there is no pair of linearly dependent rows or columns, so we have only one relation for the three vectors. The linear operator B acts on vectors in R3 as a cross product with v followed by a sign inversion of the first two components.
Such linear maps are usually referred to as pseudo-rotations or Lorentz transformations and have vital importance in hyperbolic geometry, relativity and quantum mechanics. Obtain the eigenvectors of J and the block matrix A constructed with it above.
Change each of the above parame- ters separately to see how it affects the preserved directions. What happens if you include also scaling, e. Recall the construction of the cross product in R3 and note that fixing one of the factors it is reduced to a linear map on the other, i. Then, for a fixed rotation vector the eigenvectors do not depend on t, but the eigenvalues do.
Show this dependence explicitly. Examples emerge from various fields starting with elementary Euclidean geometry and arriving at rather ad- vanced mathematical physics e. Now, let us see how this idea may be generalized to higher dimen- sions.
This splitting of the quadric into a pair of lines is referred to as degeneracy. Note that usually quadrics are considered in projective geometric perspec- tive, but here we study their basic properties from Euclidean point of view. In the generic setting the sections are ellipses or hyperbo- lae with various eccentricity, but there are also degenerate cases, in which it may be a line the plane is tangent to the cone or even a point the cusp. We shall assume a somewhat simplistic approach, which later will appear to be justified.
Namely, let us declare the three types of non-degenerate conics with their canonical equations and set a clear classification criterion. Then, we shall use our knowledge on Euclidean motions in the plane to solve the problem of obtaining this canonical form of an arbitrary quadratic polynomial in x and y.
There are three types of conics see Figure Note that one may choose only one of each pair of equations for the hyper- bola and the parabola as a canonical representative since the other is ob- tained by a quarter-turn in the plane. However, it is sometimes convenient to have such freedom of choice. The classification theorem for quadratic pla- nar curves states that they are all conic sections, i.
An analogous 3 respectively half of the minimal and maximal diameter called just axes. Moreover, all planar geometries have been classified in a similar manner as elliptic e.
Then, if conics are that important, let us study their structure into more detail. It turns out that it is not necessary to intersect cones with planes each time we would like to generate such a quadric as we may define it also as locus of points.
For instance, all points in the circle have the prop- erty that they are at the same distance R form the center. Similarly, for the ellipse The generalization of the above construction for ellipses sounds like this: the sum of the distances from the two foci F1,2 is the same for each point of the ellipse. A similar construction works for the hyperbola, but there one takes the difference, rather than the sum of distances from the two foci.
Of course, we need to determine the positions of the two foci in either case, and relate them to the canonical equation. It appears helpful to first define the eccentricity of the ellipse Now, attempt to derive the canonical equation of the ellipse With a little more effort one may prove do it! This very property holds also for the hyperbola if we consider light rays as infinite lines.
This product is part of the following series. Click on a series title to see the full list of products in the series. Examples in book are accompanied by similar practice problems that enable students to test their understanding of the material Complete solutions to the practice problems are included with the text. Each chapter ends with a set of review exercises that provide practice with all the main topics of each chapter. For a proof-oriented course, the authors have included a significant number of accessible exercises requiring proofs.
They are ordered according to difficulty. Chapter 1 Review Exercises. Chapter 2 Review Exercises. Chapter 3 Review Exercises. Chapter 4 Review Exercises. Chapter 5 Review Exercises. Chapter 6 Review Exercises. Chapter 7 Review Exercises. Pearson offers affordable and accessible purchase options to meet the needs of your students.
Connect with us to learn more. We're sorry! We don't recognize your username or password. Please try again. The work is protected by local and international copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. You have successfully signed out and will be required to sign back in should you need to download more resources.
Lawrence E. Friedberg, Illinois State University. Pearson Modern Classics are acclaimed titles at a value price. Please visit www.
0コメント