- 7.3.1: Verify thaI IS an orthogonal matrix.
- 7.3.2: Find the inverse of each of the following orthogonal matrices
- 7.3.3: Show that if A and B are orthogonal matrices. then A B IS an orthog...
- 7.3.4: Show that if A is an orthogonal matrix. then A- I is orthogonal.
- 7.3.5: Prove Theorem 7.8.
- 7.3.6: Verify Theorem 7.8 for the matrices in Exercise 2
- 7.3.7: Verify that the matrix I' in Example 3 is an orthogonal matrix and ...
- 7.3.8: Show that if A is an orthogonal matrix. then det{A) = l.
- 7.3.9: (a) Verify that the matrix is orthogonal. [ cos sin -Sin] cos (b) P...
- 7.3.10: For the orthogonal matrix verify that (Ax. Ay) = (x. y) for any vec...
- 7.3.11: Let A be an II x 1/ orthogonal matrix. and let L : R" -+ R" be the ...
- 7.3.12: A linear operator L: V ....... V. where V is an 1/- dimensional Euc...
- 7.3.13: Let L: R" -+ Rt be the linear operator performing a counterclockwis...
- 7.3.14: Let A be an 1/ x 1/ matrix and let B = 1'- 1 A I' be similar to A. ...
- 7.3.15: II/ .{ercise.l 15 rhrollgh 20. diagol/alize each gil'en malrL{ lind...
- 7.3.16: II/ .{ercise.l 15 rhrollgh 20. diagol/alize each gil'en malrL{ lind...
- 7.3.17: II/ .{ercise.l 15 rhrollgh 20. diagol/alize each gil'en malrL{ lind...
- 7.3.18: II/ .{ercise.l 15 rhrollgh 20. diagol/alize each gil'en malrL{ lind...
- 7.3.19: II/ .{ercise.l 15 rhrollgh 20. diagol/alize each gil'en malrL{ lind...
- 7.3.20: II/ .{ercise.l 15 rhrollgh 20. diagol/alize each gil'en malrL{ lind...
- 7.3.21: III 'u/Lisc.1" 2 J Ihmugh 28. diagol/alize each gil'en II/(l/ri.>:....
- 7.3.22: III 'u/Lisc.1" 2 J Ihmugh 28. diagol/alize each gil'en II/(l/ri.>:....
- 7.3.23: III 'u/Lisc.1" 2 J Ihmugh 28. diagol/alize each gil'en II/(l/ri.>:....
- 7.3.24: III 'u/Lisc.1" 2 J Ihmugh 28. diagol/alize each gil'en II/(l/ri.>:....
- 7.3.25: III 'u/Lisc.1" 2 J Ihmugh 28. diagol/alize each gil'en II/(l/ri.>:....
- 7.3.26: III 'u/Lisc.1" 2 J Ihmugh 28. diagol/alize each gil'en II/(l/ri.>:....
- 7.3.27: III 'u/Lisc.1" 2 J Ihmugh 28. diagol/alize each gil'en II/(l/ri.>:....
- 7.3.28: III 'u/Lisc.1" 2 J Ihmugh 28. diagol/alize each gil'en II/(l/ri.>:....
- 7.3.29: Prove Theorem 7.9 for the 2 x 2 case by stlldying the two possible ...
- 7.3.30: Let L: V -'> V be an oI1hogonallinear operator (see Exercise 12), w...
- 7.3.31: LetL: R2 --;. R2 bedefinedby L ( [,]) I;' ;, l [x]. ) 1;,-;,1' Show...
- 7.3.32: Let L: R2 -'> H2 be defined by L (x ) = Ax. for x in H2. where A is...
- 7.3.33: Let L. N" --;. H" be a linear operator. (a) Prove that if L is an i...
- 7.3.34: Let L: R" --+ R" be a linear operator defined by L(x ) = Ax for x i...
- 7.3.35: Let L. N" -'> H" be a linear operator and S = (v ,. V2 .. vn) an or...
- 7.3.36: Show that if AT Ay =y forallyin R" .then A T A = I".
- 7.3.37: Show that if A is an orthogonal matrix. then A T is also orthogonal.
- 7.3.38: Let A be an orthogonal matrix. Show that cA is orthogonal if and on...
- 7.3.39: Assuming that the software you use has a command for eigenvalues an...
- 7.3.40: If the answer to Exercise 39 is no. you can use the GramSchmidt pro...
Solutions for Chapter 7.3: Diagonalization of Symmetric Matrices
Full solutions for Elementary Linear Algebra with Applications | 9th Edition
ISBN: 9780132296540
Solutions for Chapter 7.3
31
4
This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.3: Diagonalization of Symmetric Matrices includes 40 full step-by-step solutions. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Since 40 problems in chapter 7.3: Diagonalization of Symmetric Matrices have been answered, more than 58249 students have viewed full step-by-step solutions from this chapter.
Key Math Terms and definitions covered in this textbook
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Cofactor Cij.
Remove row i and column j; multiply the determinant by (-I)i + j •
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Column space C (A) =
space of all combinations of the columns of A.
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Dimension of vector space
dim(V) = number of vectors in any basis for V.
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Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
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Gauss-Jordan method.
Invert A by row operations on [A I] to reach [I A-I].
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Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
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Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
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Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
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Singular matrix A.
A square matrix that has no inverse: det(A) = o.
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Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
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Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.
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Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
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Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·