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# Matrix Theory and Linear Algebra MAT 335

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This 47 page Class Notes was uploaded by Florian Watsica on Thursday October 15, 2015. The Class Notes belongs to MAT 335 at Murray State University taught by Donald Adongo in Fall. Since its upload, it has received 35 views. For similar materials see /class/223601/mat-335-murray-state-university in Mat Mathematics at Murray State University.

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Date Created: 10/15/15

Determinants 44 A First Look at Eigenvalues and Eigenvectors October 9 2009 Fixed Points A xed point ofan n X n matrix A is a vector x in R such that Axxi Problem 442 If A is an n X n matrix for what values of the scalar A if any are there non zero vectors in R such t at Ax Ax Definition 443 If A is an n X n matrix then the scalar A is called an eigenvalue of A if there is a nonzero vector x such that Ax Ax If A is an eigenvalue of A then every nonzero vector x such that Ax Ax is called an eigenvector of A corresponding to How do we find the eigenvalues Theorem 444 If A is an n x n matrix and A is a scalar then the following are equivalent 1 A is an eigenvalue of A 2 A is a solution of the equation detAI 7 A O 3 The linear system AI 7 Ax O has nontrivial solutions Example Find the eigenvalues of the matrix Example Find the eigenvalues and corresponding eigenvectors of the matrix 2 73 1 O 73 2 O O 1 12 O O O Eigenvalues of Triangular Matrices If A is a triangular matrix upper triangular lower triangular or diagonal then the eigenvalues of A are the entries on the main diagonal of A Eigenvalues of Powers of a Matrix Theorem 446 If A is an eigenvalue of a matrix A and x is a corresponding eigenvector and if k is any positive integer then M is an eigenvalue of Ak and x is a corresponding eigenvector Theorem 447 lfA is an n X n matrix then the following statements are equivalent a The reduced row echelon form ofA is In b A is expressible as a product of elementary matrices c A is invertible d Ax 0 has only the trivial solution e Ax b is consistent for every vector b in R f Ax b has exactly one solution for every vector b in R g The column vectors ofA are linearly independent h The row vectors ofA are linearly independent i detA y 0 j A O is not an eigenvalue ofA If A is an n X n matrix then the expanded form of the determinant detl 7 A is a polynomial of degree n that is detl 7 A A c1 1 cn1 cni Definition Characteristic Polynomial The polynomial p A clAn l cn1 c is called the characteristic polynomial of A Theorem 4412 If A is an n X n matrix with eigenvalues A12Mn repeated according to multiplicity then 1 detA AlAzwAn 2 trA A1 A2 An Consider a 2 X 2 matrix A Dimension and Structure 77 The Projection Theorem and Its Implications November 9 2009 Orthogonal Projections Onto Lines in R2 Let a be a nonzero vector in R2 We would like to compute the orthogonal projection of a vector x onto the line W spana y The vector x can be expressed as x x1 xz where gt x1 is the orthogonal projection ofx onto W gt xz x 7 x1 is the orthogonal projection onto the line through the origin that is perpendicular to W Note that gt the vector x1 is a scalar multiple ofa say x1 ka gt the vector xz x 7 x1 x 7 ka is orthogonal to a Hence x 7 ka a O thatis x a 7 ka a Oi Solvefork xaikHaH2 0 k Hence We denote the orthogonal projection of x onto the line spana by xa prOJ x far a iiaii2 Example Find the orthogonal projection of x 732 on the line x3y O Orthogonal Projections Onto Lines Through the Origin of Rn Theorem 771 If a is a nonzero vector in R then every vector x in R can be expressed in exactly one way as X X1 X2 where x1 is a scalar multiple ofa and xz is orthogonal to a and hence to x1 The vectors x1 and xz are given by the formulas x a 2 Hall xa x1 a and xzx772a llall Definition 772 If a is a nonzero vector in R and if x is any vector in R then the orthogonal projection of x onto spana is denoted by projax and is defined as x a pI OJaX War gt The vector projax is also called the vector component of x along a gt The vector x 7 projax is called the vector component of x orthogonal to Example Find the vector components of x along a and orthogonal to a gt x 201 a 123 x 5 0 737 a 21 7171 Example Find the length of the orthogonal projection of x on a gt x 4 751 a 224 x 5 7371 a 71071 Hint It is easier to use llprojaxll Matrices and Matrix Algebra 36 Matrices with Special Forms September 21 2009 Definition A diagonal matrix is a square matrix in which all entries off the main diagonal are zero Thus a general n X n diagonal matrix has the form d1 0 O 0 d2 O O O dn A diagonal matrix is invertible ifand only ifall of its diagonal entries are non zero The inverse is given by dil 0 0 1 0 dig 0 D 0 0 d7 If k is a positive integer then the kth power of a diagonal matrix is cllk 0 0 Dk 0 d 0 0 0 d The result is also true if k lt O and D is invertible Example 1 Consider the diagonal matrix 100 A07300 002 Then 1 100 100 A 10730 07 0 002 00 100515 00 1 A57 0 30 0735 0 0 002 0 025 0 O O 7243 O O 32 Triangular Matrices gt A square matrix in which all entries above the main diagonal are zero is called lower triangular gt A square matrix in which all entries below the main diagonal are zero is called upper triangular gt A matrix that is either upper triangular or lower triangular is called triangular Example 2 General 4 x 4 triangular matrices have the forms 31 1 312 313 314 311 0 0 0 0 322 323 324 321 322 0 0 0 0 333 334 331 332 333 0 O O 0 344 341 342 343 344 Theorem 361 a The transpose ofa lower triangular matrix is upper triangular and t e transpose of an upper triangular matrix is lower triangular b A product of lower triangular matrices is lower triangular and a product of upper triangular matrices is upper triangular c A triangular matrix is invertible if and only if its diagonal entries are all nonzero d The inverse of an invertible lower triangular matrix is lower triangular and the inverse ofan invertible upper triangular matrix is upper triangular A square matrix A is called symmetric if AT A and skew symmetric if AT 7A Examples Symmetric l7 ilv Skewsymmetric OOO Theorem 362 IfA and B are symmetric matrices With the same size and ifk is any scalar then a AT is symmetric b A B and A 7 B are symmetric c kA is symmetric Theorem 363 The product of two symmetric matrices is symmetric if and only if the matrices commute Theorem 364 Invertibility of symmetric matrices lfA is an invertible symmetric matrix then A 1 is symmetric Proof Must show that A 1 is equal to its transpose 1 Theorem 365 lfA is a square matrix then the matrices A AAT and ATA are either all invertible or all singular Definition A square matrix A with the property that Ak O for some positive integer k is said to be nilpotent The smallest positive power for which Ak O is called the index of nilpotency Theorem 366 IfA is a square matrix and A is nilpotent then the matrix 7 A is invertible and liA 1IAA2mAk 1i Example 7 Consider the matrix A OOO OOM OLDH l l Dimension and Structure 71 Basis and Dimension October 26 2009 Definition 711 A set of vectors in a subspace V of R is said to be a basis for V if it is linearly independent and spans V Recall gt linear independence gt span Note that e17 e2 Me is a basis for R It is called the standard basis for R Example Is the set of vectors 17 7121713 a basis for R2 Example Find a basis for the null space of the matrix 1 2 O 2 3 6 O 6 72 75 73 18 Theorem 713 If V is a nonzero subspace of R then there exists a basis for V that has at most n vectors Recall Theorem 348 A set with more than n vectors in R is linearly dependent Theorem 714 All bases for a nonzero subspace of R have the same number of vectors Definition 715 If V is a nonzero subspace of R then the dimension of V written dimV is defined to be the number of vectors in a basis for V In addition we define the zero subspace to have dimension 0 Recall that for 1372 020 A7267572473 00510015 2608418 Theorem 713 If a is a nonzero vector in R then dimai n71 Note aix6R xa0 that is ai is the set of all vectors that are orthogonal to a Example Consider a 111 Find a basis for ai Example Are the vectors 111 110 and 005 a basis for R3 Determinants 43 Cramer39s Rule Formula for A l Applications of Determinants October 7 2009 Definition 432 If A is an n x n matrix and Cu is the cofactor of 3 then the matrix C11 C12 Cln C21 C22 Czn C CL C in Cm is called the matrix of cofactors from A The transpose of this matrix is called the adjoint or sometimes the adjugate of A and is denoted by adjA Example Find the adjoint of Example Find the adjoint of Theorem 433 IfA is an invertible matrix then 1 71 7 A 7 detA adjA Proof Let B declAadjA The we need to show that AB BA I Let 211 212 31m C11 Q1 Cm 39 C12 Q2 Cn2 A 21 22 2m ad A 3 Cm Qn Cnn 2m 2n2 3m Consider A adjAU Example Find A 1 using the adjoint method if ii ii Example Find A 1 using the adjoint method if 172 71 O 01 174 3 A Theorem 434 Cramer39s Rule lfo b is a linear system ofn equations in n unknowns then the system has a unique solution if and only if detA y O in Which case the solution is 7 detA1 7 detA2 7 detAn Xl detA XZ detA X detA Where A1 is the matrix that results When the jth column ofA is replaced by b Example Use Cramer39s rule to solve the system Xy X y Example Use Cramer39s rule to solve the system x y 7 2X7 y ix2y 2 z 7 72 z 5 s 22 7 1 Theorem 436 Suppose that a triangle in the xy plane has vertices P1 X17 yl P2 X2y2 and l33x37 yg and that the labeling is such that the triangle is traversed counterclockwise from P1 to P2 to P3 Then the area of the triangle is given by 1 X1 Y1 1 area APl P2 P3 E X2 yz 1 X3 Y3 1 Example Find the area of the triangle with vertices A27 0 837 4 and C712 Dimension and Structure 73 The Fundamental Spaces of a Matrix October 30 2009 Three spaces associated with an m X n matrix A are 1 the row space of A denoted by rowA It is the subspace of R that is spanned by the row vectors of 2 the column space of A denoted by colA It is the subspace of R that is spanned by the column vectors 0 the null space of A denoted by nu11A It is the solution space of Ax 0 Recall it is a subspace of R 5 Definition 731 The dimension of the row space of a matrix A is called ther rank of A and is denoted by rankA and the dimension of the null space of A is called the nullity of A and is denoted by nullityA Example Describe the column spaces for a A C1 1 J The whole space R2 b B i i Aline through the origin containing 12 7 1 2 3 2 cC7OO4AllofR 1 O 1 0 d D 2 3 4 1 The plane in R3 passing through the origin 4 3 6 1 containing 124 and 033 Example Describe the row spaces for 1 O 1 O a A 2 3 4 1 A basis for rowA is 4 3 6 1 1 70T70717 7T b B O O 4 A basis for rOWB is B 120T001T Defintion Orthogonal Complement Let M be a subspace of R The orthogonal complement of M denoted by Hi is the set of all vectors v in R such that v L u for every vector u in M Theorem Let M be a subspace o R Then 1 Mi is a subspace ofR 2 The intersection of the subspaces M and ML is the zero subspace that is Mi Il 0 3 NHL u Example Find the orthogonal complement in an xyzcoordinate system of the set 5 177271737 7775 Hint Show that a general solution of the resulting system is X 3139 y 2t 2 139 Si is the line through the origin that is parallel to the vector w 321 Theorem 735 If A is an m X n matrix then the row space of A and the null space of A are orthogonal complements Theorem 736 If A is an m X n matrix then the column space of A and the null space of AT are orthogonal complements Theorem 738 If A and B are matrices with the same number of columns then the following statements are equivalent a A and B have the same row space b A and B have the same nu space c The row vectors of A are linear combinations of the row vectors of B and converse y

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