History and Applications of Matrices

History and Applications of MatricesMatrices light upon many applications at current time and very useful to us. Physics makes use of matrices in various domains, for example in geometrical optics and matrix mechanics the latter led to studying in more detail matrices with an infinite number of forms and tugboats. Graph theory uses matrices to keep line of distances between pairs of vertices in a graph. Computer graphics uses matrices to project 3-dimensional space onto a 2-dimensional screen.Example of applicationA message is converted into numeric conformity according to some scheme. The easiest scheme is to let space=0, A=1, B=2, , Y=25, and Z=26. For example, the message Red Rum would become 18, 5, 4, 0, 18, 21, 13.This data was located into matrix form. The size of the matrix depends on the size of the encryption key. Lets say that our encryption matrix (encoding matrix) is a 22 matrix. Since I submit seven pieces of data, I would place that into a 42 matrix and fill the last spot with a space to make the matrix complete. Lets shoot the breeze the buffer, unencrypted data matrix A.thither is an invertible matrix which is called the encryption matrix or the encoding matrix. Well call it matrix B. Since this matrix needs to be invertible, it must be square.This could truly be anything, its up to the person encrypting the matrix. Ill use this matrix.The unencrypted data is then multiplied by our encoding matrix. The result of this multiplication is the matrix containing the encrypted data. Well call it matrix X.The message that you would pass on to the other person is the the stream of numbers 67, -21, 16, -8, 51, 27, 52, -26.Decryption ProcessPlace the encrypted stream of numbers that represents an encrypted message into a matrix.Multiply by the decoding matrix. The decoding matrix is the inverse of the encoding matrix.Convert the matrix into a stream of numbers.Conver the numbers into the text of the received message.DETERMINANTSThe epitope of a matrix A is pertaind det(A), or without parentheses det A. An alternative notation, used for compactness, especially in the case where the matrix entries are written out in full, is to denote the determinant of a matrix by surrounding the matrix entries by vertical bars instead of the usual brackets or parentheses.For a fixed nonnegative integer n, there is a unique determinant function for the n-n matrices over any commutative ring R. In particular, this unique function exists when R is the field of real or complex numbers.For any square matrix of rules of order 2, we mystify found a necessary and sufficient condition for invertibility. Indeed, trade the matrixExample. EvaluateLet us transform this matrix into a triangular hotshot through elementary operations. We will keep the first row and add to the second one the first multiplied by . We getUsing the Property 2, we getTherefore, we havewhich one may check easily.EIGEN value AND EIGEN VECTORSIn mathematics, eigenvalue, eig envector, and eigenspace are related concepts in the field of additive algebra. The prefix eigen- is adopted from the German word eigen for innate, idiosyncratic, own. Linear algebra studies linear transformations, which are represented by matrices acting on vectors. Eigenvalues, eigenvectors and eigenspaces are properties of a matrix. They are computed by a method described below, give important information about the matrix, and can be used in matrix factorization. They have applications in areas of applied mathematics as diverse as economics and quantum mechanics.In general, a matrix acts on a vector by changing both its magnitude and its direction. However, a matrix may act on certain vectors by changing alone their magnitude, and leaving their direction same(predicate) (or possibly reversing it). These vectors are the eigenvectors of the matrix. A matrix acts on an eigenvector by multiplying its magnitude by a factor, which is positive if its direction is unchanged and negati ve if its direction is reversed. This factor is the eigenvalue associated with that eigenvector. An eigenspace is the stack of all eigenvectors that have the same eigenvalue, together with the zero vector.These concepts are formally defined in the language of matrices and linear transformations. Formally, if A is a linear transformation, a non-null vector x is an eigenvector of A if there is a scalar such thatThe scalar is said to be an eigenvalue of A corresponding to the eigenvector x.Eigenvalues and Eigenvectors An IntroductionThe eigenvalue problem is a problem of considerable theoretical interest and wide-ranging application. For example, this problem is crucial in solving systems of differential equations, analyzing existence growth models, and calculating powers of matrices (in order to define the exponential matrix). Other areas such as physics, sociology, biology, economics and statistics have focused considerable attention on eigenvalues and eigenvectors-their applicat ions and their computations. sooner we give the formal definition, let us introduce these concepts on an example.Example.Consider the matrixConsider the three column matricesWe haveIn other words, we haveNext consider the matrix P for which the columns are C1, C2, and C3, i.e.,We have det(P) = 84. So this matrix is invertible. Easy calculations giveNext we evaluate the matrix P-1AP. We leave the expound to the reader to check that we haveIn other words, we haveUsing the matrix multiplication, we obtainwhich implies that A is similar to a diagonal matrix. In particular, we havefor . Note that it is almost impossible to find A75 directly from the original form of A.This example is so rich of conclusions that many questions impose themselves in a subjective way. For example, given a square matrix A, how do we find column matrices which have similar behaviors as the above ones? In other words, how do we find these column matrices which will help find the invertible matrix P such that P-1AP is a diagonal matrix?From now on, we will call column matrices vectors. So the above column matrices C1, C2, and C3 are now vectors. We have the following definition.Definition. Let A be a square matrix. A non-zero vector C is called an eigenvector of A if and only if there exists a number (real or complex) such thatIf such a number exists, it is called an eigenvalue of A. The vector C is called eigenvector associated to the eigenvalue .Remark. The eigenvector C must be non-zero since we havefor any number .Example. Consider the matrixWe have seen thatwhereSo C1 is an eigenvector of A associated to the eigenvalue 0. C2 is an eigenvector of A associated to the eigenvalue -4 while C3 is an eigenvector of A associated to the eigenvalue 3.It may be interesting to distinguish whether we found all the eigenvalues of A in the above example. In the next page, we will discuss this question as well as how to find the eigenvalues of a square matrix.PROOFS OF PROPERTIES OF EIGEN VALUESP ROPERTY 1Inverse of a matrix A exists if and only if zero is not an eigenvalue of ASuppose A is a square matrix. Then A is singular if and only if =0 is an eigenvalue of A.Proof We have the following equivalencesA is singularthere exists x0, Ax=0there exists x0, Ax=0x=0 is an eigenvalue of ASince SINGULAR matrix A has eigenvalue and the inverse of a singular matrix does not exist this implies that for a matrix to be invertible its eigenvalues must be non-zero.PROPERTY-2Eigenvalues of a matrix are real or complex conjugates in pairsSuppose A is a square matrix with real entries and x is an eigenvector of A for theeigenvalue . Then x is an eigenvector of A for the eigenvalue . -ProofAx =Ax=Ax=x=xA has real entries x eigenvector of ASuppose A is an m-n matrix and B is an n-p matrix. Then AB=AB. -Proof To obtain this matrix equality, we will work entry-by-entry. For 1im, 1jp,ABij =ABij =nk=1AikBkj =nk=1AikBkj =nk=1AikBkj =nk=1AikBkj =ABijAPPLICATION OF EIGEN VALUES IN FACIAL RECOGNITION How does it work?The task of facial recogniton is discriminating scuttlebutt signals ( consider data) into several classes (persons). The input signals are highly noisy (e.g. the noise is caused by differing lighting conditions, pose etc.), yet the input ikons are not completely haphazard and in spite of their differences there are patterns which occur in any input signal. Such patterns, which can be observed in all signals could be in the domain of facial recognition the presence of some objects (eyes, nose, mouth) in any face as well as relative distances between these objects. These mark features are called eigenfaces in the facial recognition domain (or principal components generally). They can be extracted out of original image data by means of a mathematical tool called Principal Component Analysis (PCA).By means of PCA one can transform each original image of the training set into a corresponding eigenface. An important feature of PCA is that one can reconstruct reconst ruct any original image from the training set by cartel the eigenfaces. Remember that eigenfaces are nothing less than characteristic features of the faces. Therefore one could say that the original face image can be reconstructed from eigenfaces if one adds up all the eigenfaces (features) in the right proportion. Each eigenface represents only certain features of the face, which may or may not be present in the original image. If the feature is present in the original image to a higher degree, the share of the corresponding eigenface in the sum of the eigenfaces should be greater. If, contrary, the particular feature is not (or almost not) present in the original image, then the corresponding eigenface should contribute a smaller (or not at all) part to the sum of eigenfaces. So, in order to reconstruct the original image from the eigenfaces, one has to build a kind of weighted sum of all eigenfaces. That is, the reconstructed original image is equal to a sum of all eigenfaces, w ith each eigenface having a certain weight. This weight specifies, to what degree the specific feature (eigenface) is present in the original image.If one uses all the eigenfaces extracted from original images, one can reconstruct the original images from the eigenfaces exactly. But one can also use only a part of the eigenfaces. Then the reconstructed image is an approximation of the original image. However, one can ensure that losses due to omitting some of the eigenfaces can be minimized. This happens by choosing only the most important features (eigenfaces). failure of eigenfaces is necessary due to scarcity of computational resources.How does this relate to facial recognition? The clue is that it is possible not only to extract the face from eigenfaces given a set of weights, but also to go the opposite way. This opposite way would be to extract the weights from eigenfaces and the face to be recognized. These weights tell nothing less, as the nub by which the face in question differs from typical faces represented by the eigenfaces. Therefore, using this weights one can determine two important thingsDetermine, if the image in question is a face at all. In the case the weights of the image differ too much from the weights of face images (i.e. images, from which we know for sure that they are faces), the image probably is not a face.Similar faces (images) possess similar features (eigenfaces) to similar degrees (weights). If one extracts weights from all the images available, the images could be grouped to clusters. That is, all images having similar weights are likely to be similar faces.