Section 7.3: SYMMETRIC MATRICES AND ORTHOGONAL DIAGONALIZATION

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1 Section 7.3: SYMMETRIC MATRICES AND ORTHOGONAL DIAGONALIZATION When you are done with your homework you should be able to Recognize, and apply properties of, symmetric matrices Recognize, and apply properties of, orthogonal matrices Find an orthogonal matrix P that orthogonally diagonalizes a symmetric matrix A SYMMETRIC MATRICES Symmetric matrices arise more often in than any other major class of matrices. The theory depends on both and. For most matrices, you need to go through most of the diagonalization to ascertain whether a matrix is. We learned about one exception, a matrix, which has entries on the main. Another type of matrix which is guaranteed to be is a matrix. DEFINITION OF SYMMETRIC MATRIX A square matrix A is when it is equal to its :. Example 1: Determine which of the matrices below are symmetric. A , B , C D 1 2 3, CREATED BY SHANNON MARTIN GRACEY 205

2 Example 2: Using the diagonalization process, determine if A is diagonalizable. If so, diagonalize the matrix A. A CREATED BY SHANNON MARTIN GRACEY 206

3 THEOREM 7.7: PROPERTIES OF SYMMETRIC MATRICES If A is an n n symmetric matrix, then the following properties are true. 1. A is. 2. All of A are. 3. If is an of A with multiplicity, then has linearly eigenvectors. That is, the of has dimension. Proof of Property 1 (for a 2 x 2 symmetric matrix): CREATED BY SHANNON MARTIN GRACEY 207

4 Example 3: Prove that the symmetric matrix is diagonalizable. a a a A a a a a a a CREATED BY SHANNON MARTIN GRACEY 208

5 Example 4: Find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. A CREATED BY SHANNON MARTIN GRACEY 209

6 DEFINITION OF AN ORTHOGONAL MATRIX A square matrix P is when it is and when. THEOREM 7.8: PROPERTY OF ORTHOGONAL MATRICES An n n matrix P is orthogonal if and only if its vectors form an set. Proof: CREATED BY SHANNON MARTIN GRACEY 210

7 Example 5: Determine whether the matrix is orthogonal. If the matrix is orthogonal, then show that the column vectors of the matrix form an orthonormal set. A CREATED BY SHANNON MARTIN GRACEY 211

8 THEOREM 7.9: PROPERTY OF SYMMETRIC MATRICES Let A be an n n symmetric matrix. If 1 and 2 are eigenvalues of A, then their corresponding x 1 and x 2 are. Proof: CREATED BY SHANNON MARTIN GRACEY 212

9 THEOREM 7.10: FUNDAMENTAL THEOREM OF SYMMETRIC MATRICES Let A be an n n matrix. Then A is and has eigenvalues if and only if A is. Proof: CREATED BY SHANNON MARTIN GRACEY 213

10 STEPS FOR DIAGONALIZING A SYMMETRIC MATRIX Let A be an n n symmetric matrix. 1. Find all of A and determine the of each. 2. For eigenvalue of multiplicity, find a eigenvector. That is, find any and then it. 3. For eigenvalue of multiplicity, find a set of eigenvectors. If this set is not, apply the process. 4. The results of steps 2 and 3 produce an set of eigenvectors. Use these eigenvectors to form the of. The matrix will be. The main entries of are the of. CREATED BY SHANNON MARTIN GRACEY 214

11 T Example 5: Find a matrix P such that P AP orthogonally diagonalizes A. Verify T that P AP gives the proper diagonal form. A CREATED BY SHANNON MARTIN GRACEY 215

12 Example 6: Prove that if a symmetric matrix A has only one eigenvalue, then A I. CREATED BY SHANNON MARTIN GRACEY 216

13 Section 7.4: APPLICATIONS OF EIGENVALUES AND EIGENVECTORS When you are done with your homework you should be able to Find the matrix of a quadratic form and use the Principal Axes Theorem to perform a rotation of axes for a conic and a quadric QUADRATIC FORMS Every conic section in the xy-plane can be written as : If the equation of the conic has no xy-term ( ), then the axes of the graphs are parallel to the coordinate axes. For second-degree equations that have an xy-term, it is helpful to first perform a of a c axes that eliminates the xy-term. The required rotation angle is cot 2. b 2 With this rotation, the standard basis for R, is rotated to form the new basis. CREATED BY SHANNON MARTIN GRACEY 217

14 Example 1: Find the coordinates of a point x, y in cos,sin, sin, cos B. 2 R relative to the basis ROTATION OF AXES The general second-degree equation 2 2 written in the form 2 2 ax bxy cy dx ey f 0 can be a x c y d x ey f 0 by rotating the coordinate axes counterclockwise through the angle, where is defined by a c cot 2. The coefficients of the new equation are obtained from the b substitutions x xcos ysin and y xsin ycos. CREATED BY SHANNON MARTIN GRACEY 218

15 Example 2: Perform a rotation of axes to eliminate the xy-terms in 2 2 5x 6xy 5 y 14 2x 2 2 y Sketch the graph of the resulting equation. CREATED BY SHANNON MARTIN GRACEY 219

16 and can be used to solve the rotation of axes problem. It turns out that the coefficients a and c are eigenvalues of the matrix The expression is called the form associated with the quadratic equation and the matrix is called the of the form. Note that is. Moreover, will be if and only if its corresponding quadratic form has no term. Example 3: Find the matrix of quadratic form associated with each quadratic equation. a. x 4 y b x 6xy 5 y 14 2x 2 2 y 18 0 CREATED BY SHANNON MARTIN GRACEY 220

17 Now, let s check out how to use the matrix of quadratic form to perform a rotation of axes. Let X x y. Then the quadratic expression matrix form as follows: 2 2 ax bxy cy dx ey f can be written in If, then no is necessary. But if, then because is symmetric, you may conclude that there exists an matrix such that is diagonal. So, if you let then it follows that, and The choice of must be made with care. Since is orthogonal, its determinant will be. If P is chosen so that P 1, then P will be of CREATED BY SHANNON MARTIN GRACEY 221

18 the form where gives the angle of rotation of the conic measured from the x-axis to the positive x -axis. PRINCIPAL AXES THEOREM For a conic whose equation is 2 2 ax bxy cy dx ey f 0, the rotation given by eliminates the xy-term when P is an orthogonal matrix, with P 1, that diagonalizes A. That is where 1 and 2 are eigenvalues of A. The equation of the rotated conic is given by CREATED BY SHANNON MARTIN GRACEY 222

19 Example 4: Use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system x 6xy 5 y 14 2x 2 2 y 18 0 CREATED BY SHANNON MARTIN GRACEY 223

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