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1 Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant. For example the areas of volumes of certain geometric figures can be found by calculating the determinant of the matrix made up of the vectors that form these geometric figures. We will also use the determinant to introduce a special value called the eigenvalue of the square matrix. The eigenvectors associated with the eigenvalue have the special property that when multiplied by the matrix they produce a parallel vector. Introduction to Determinants Any square nxn matrix A has a determinant. The notation is A or det(a). a 1x1 matrix A= [ a 11 ] has a determinant determined by its only term det(a) = a 11 a b a 2x2 matrix B= c d has a determinant determined by det(b) = ad bc c11 c12 c13 a 3x3 matrix C= c21 c22 c 23 has a determinant determined by c31 c32 c 33 det(c) = c 11 c 22 c 33 + c 12 c 23 c 31 + c 13 c 21 c 32 c 13 c 22 c 31 c 11 c 23 c 32 c 12 c 21 c 33 there is a pattern that makes this easy to memorize, its often called the basket weaving technique. c c c c c c c c c c c c c c c Ex: Calculate the determinants of the following matrices 7 5 A = 8 6 B= and C=

2 Applications of Determinants: Areas and Volumes 2 v= v, v w= w, w in R that are not parallel and share a starting vertex. Let [ ] and [ ] Then the area of the parallelogram formed by v and w if given by the determinant of v1 v2 w1 w 2 (take the absolute value since area should be non-negative) w v Let v= [ v, v, v ] and w= [ w, w, w ] and z= [ z, z, z ] in 3 R that are not coplanar and they all share the same initial point. Then the volume of the parallelepiped form by v, w, and z is given by the determinant of (again absolute value because volume) v1 v2 v3 w1 w2 w 3 z1 z2 z 3 z w v What happens when the vectors are parallel or coplanar? Calculating Determinants by Coplanar Expansion There is another method of finding determinants using elements of a row multiplied by determinants of square matrices created by certain elements in the other rows. Let A be a nxn matrix with n 2. The (i,j) th sub-matrix A ij of the matrix A is the (n)x(n) matrix obtained by deleting the i th row of A and the j th column of A. The (i,j) th minor, denoted M ij, is the determinant of the submatrix A ij of A. Let A be a nxn matrix with n 2. The (i,j) th cofactor of A, denoted C ij, is () i+j multiplied by the (i,j) th minor of A. In other words C ij = () i+j M ij So given an 3x3 matrix there are nine submatrices, nine minors, and nine cofactors. Ex: Find all nine submatrices, minors, and cofactors of A=

3 The Formal Definition of the Determinant Let A be an nxn matrix. The determinant of A, denoted det(a), is defined as follows: If n = 1, then det(a) = a 11 If n >1, then det(a) = a n1 C n1 + a n2 C n2 + a n3 C n3 + + a nn C nn When we calculate the determinant using the sum of cofactors, we call the procedure the cofactor expansion Ex: Given A= calculate the determinant by cofactor expansion 3 ways We could also define the determinant of a nxn matrix A with n >1 as det(a) = a 1n C 1n + a 2n C 2n + a 3n C 3n + + a nn C nn Choose the row/column with the most zeros to simplify the computation since any of them with produce the determinant. Ex: Calculate the determinant by cofactor expansion for the following A. A= B. B= Determinants by Row Reduction Determinant of Upper Triangular Matrices: Let A be an upper triangular nxn matrix then, det(a) = a 11 a 22 a 33.a nn (product of diagonal entries) Ex: calculate det(a) if A= What effect does row reduction have on the determinant? Let A be a nxn matrix with the det(a)=p and let c be a scalar, then: If the row operation is R i R j then the determinant becomes P If the row operation if R i = cr i the the determinant becomes cp If the row operation if R i = cr j +r i then the determinant stays the same Ex: Use row reduction to calculate the determinant of A= Theorem: A nxn matrix A is nonsingular iff det(a) 0 Corollary: Let A be an nxn matrix, then rank(a)=0 iff det(a) 0 Assume A is a nxn matrix, then the following are equivalent

4 A is singular (A DNE) rank(a) n det(a)=0 AX = 0 has a nontrivial solution AX = B does not have a unique solution A is nonsingular (A exists) rank(a)=n det(a) 0 AX = 0 has only the trivial solution AX = B has a unique solution Further Properties of the Determinant In this section we will discuss the determinant of a matrix product, the determinant of a transpose, the determinant and the inverse, and finish with Cramer s Rule to solve systems using only the determinants. If A and B are both nxn matrices then, det(ab) = det(a)*det(b) If A is a nxn matrix with A 1 as its inverse then, det(a) = 1 det( A ) If A is an nxn matrix then, det(a)=det(a T ) The Adjoint Matrix Let A be a nxn matrix with n 2 then, the adjoint, denoted adj(a), of A is the nxn matrix whose ij th entry is the ji th cofactor C ji of A. The adjoint is the transpose of cofactors matrix. Watch the order. C11 C21... Cn 1 adj( A) = C12 C22... C n2 C1 n... C nn Ex: Find the adjoint of A= If A is a nxn matrix with adjoint, adj(a) then: A adj( A) = adj( A) A= I n If a is a nxn invertible matrix, then A adj( A) 1 1 = Cramer s Rule Cramer s Rule, named for Gabriel Cramer ( ), uses determinants to a solve a system of n linear equations in n variables. This rule only applies to systems with unique solutions.

5 Cramer s Rule : Let AX = B represent a system of n linear equations in n variables such that det(a) 0 then the solu on to the system is det( A1 ) det( A2 ) det( An ) x1 = x2 = xn = where the i th column of A i is the column of constants in the system of equations, (B). 5x 3y 10z= 9 Ex: solve by Cramer Rule 2x+ 2y 3z= 4 3x y + 5z = 1 Eigenvalues and Diagonalization We will define eigenvalues and eigenvectors for matrices in order to find when possible the diagonal form of a square matrix. Let A be a nxn matrix. A real number λ is an eigenvalue of A iff there exists any nonzero vector X for which AX = λx. Also, any nonzero vector X for which AX = λx is called an eigenvector corresponding to the eigenvalue λ. Let A be a nxn matrix and λ be an eigenvalue of A then, the set Eλ { xax λx} called the eigenspace corresponding to the eigenvalue λ. = = is The Characteristic Polynomial of a Matrix A We need procedure to determine all eigenvalues and eigenvectors of an nxn matrix A. If X is an eigenvector for A corresponding to the eigenvalue λ then AX = λx = λi n X or (λi n A)X = 0 Therefore X is a nontrivial solution to the homogeneous system whose coefficient matrix is (λi n A). Let A be a nxn matrix and λ be a real number then λ is an eigenvalue of A iff det(λi n A) = 0 The eigenvectors corresponding to λ are the nontrivial solutions X to the homogeneous system (λi n A)X = 0. The eigenspace E λ corresponding to λ us the set of all nontrivial solutions to (λi n A)X = 0 Let A be a nxn matrix then the characteristic polynomial of A is the polynomial given by P A (λ) = det (λi n A) The real roots of the characteristic polynomial are the eigenvalues of A.

6 Procedure to find Eigenvalues, Eigenvectors, and Eigenspace: 1. State the characteristic polynomial by solving the determinant P A (λ) = det (λi n A) 2. The values λ 1, λ 2, λ k that are the roots to P A (λ) are the eigenvalues 3. Set up the homogeneous system (λi n A)X = 0 where the coefficient matrices are (λi n A) for each λ 1, λ 2, λ k. The nontrivial solution(s) to this system is the eigenvector corresponding to each eigenvalue. 4. All linear combinations of the eigenvectors for each λ 1, λ 2, λ k make up the eigenspace corresponding to each eigenvalue and eigenvector. State E λk. Ex: Find all the eigenvalues of the given matrix then state the eigenvectors associated with each eigenvalue and the corresponding eigenspace a. A= b. B= Diagonalization We now seek to take a square matrix and turn it into a representative diagonal matrix by use of eigenvalues and eigenvectors. Realize if we can replace a matrix with a diagonal matrix then we will greatly simplify any computations involving the original matrix. Therefore our next goal is to present a formal method for using eigenvalues and eigenvectors to find a diagonal form of a square matrix, provided it exists. A matrix A is similar to a matrix A if there exists some nonsingular matrix P such that A = P AP. Properties of Similar Matrices Let A, B and C be nxn matrices. then the following properties are true. 1. A is similar to A 2. If A is similar to B, then B is similar to A. 3. If A is similar to B and B is similar to C then A is similar to C. 4. If A and B are similar they have the same eigenvalues. An nxn matrix A is diagonalizable when A is similar to a diagonal matrix. That is, A is diagonalizable when there exists an invertible matrix P such that D = P AP where D is a diagonal matrix. An nxn matrix A is diagonalizable iff it has n linearly independent eigenvectors. Let A and P be nxn matrices such that each column of P is an eigenvector of A. If P is nonsingular then D = P AP is a diagonal matrix similar to the matrix A. The i th main diagonal entry d ii of D is the eigenvalue for the eigenvector forming the i th column of P.

7 Notice in the two examples we did for finding eigenvalues and eigenvectors, we see if we form P in either case, then P doesn t exist and hence, the original matrices are not diagonalizable. Method for Diagonalizing a nxn matrix A (if possible) 1. Calculate the characteristic polynomial and find all the real roots, λ 1, λ 2, λ k. In other words find all λ such that if P(λ) = det (λi n A) then P(λ) = For each eigenvalue λ m in λ 1, λ 2, λ k we find the corresponding eigenvectors. If there are n λ s that are distinct, then A is diagonalizable and if there are less than n λ s, then A is not diagonalizable. 3. If there are n λ s that are distinct then there are n eigenvectors. Form a matrix P such that the columns of P are eigenvectors and find P. 4. Check your work by checking that P AP is a diagonal matrix D whose main diagonal entries are the corresponding eigenvalues for the eigenvectors that formed that column Ex: Find the diagonal matrix D such that D = P AP for A=

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