3(a).
(i). If a nxn real matrix equals its transpose,i.e. if A = AT, then A is said to be a symmetric matrix.
(ii). A real orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors.
(iii). A real nxn matrix A is said to be diagonalizable, if there exists a nxn diagonal matrix D and a nxn invertible matrix P such that A = PDP-1.
(iv). A real nxn matrix A is said to be orthogonally diagonalizable if there is an orthogonal matrix S such that S-1AS is a nxn diagonal matrix.
(b). Let A be a real nxn matrix. Then, as per the Spectral theorem, A is orthogonally diagonalizable if and only if A is symmetric.
(c ). The the solutions to its characteristic equation det(A-?I2) = 0 or, ?2-?-42 = 0 or, (?-7)(?+6) = 0. Thus, the eigenvalues of A are 7 and -6. The eigenvector of A associated with its eigenvalue 7 is solution to the equation(A-7I3)X = 0. The RREF of A-7I3 is
1 |
-3/2 |
0 |
0 |
Now, if X = (x,y)T, then the equation (A-7I3)X = 0 is equivalent to x-3y/2 = 0 or, x = 3y/2. Then X = (3y/2,y)T = (y/2)(3,2)T. Therefore,the eigenvector of A associated with its eigenvalue 7 is v1 = (3,2)T. Similarly, the eigenvector of A associated with its eigenvalue -6 is v2 = (-2,3)T.Further, v1.v2 =(3,2)T . (-2,3)T = -6+6 = 0. Hence the eigenvectors v1 and v2 of A associated with its eigenvalues 7 and -6 are orthogonal.
(d). It may be observed that AT = A i.e.tric. Hence, by the Spectral theorem, A is orthogonally diagonalizable.
3, (a) [5 marks] what does it mean for A E Rnxn to be (i) symmetric?...
8. Find a symmetric 3 x 3 matrix with eigenvalues 11, 12 , and , 13 and corresponding orthogonal eigenvectors vi , V2 , and V3 1 11 = 1, 12 = 2, 13 = 3, vi -=[:)--[:)--[;)] 1
1. Let W CR denote the subspace having basis {u, uz), where (5 marks) (a) Apply the Gram-Schmidt algorithm to the basis {uj, uz to obtain an orthogonal basis {V1, V2}. (b) Show that orthogonal projection onto W is represented by the matrix [1/2 0 1/27 Pw = 0 1 0 (1/2 0 1/2) (c) Explain why V1, V2 and v1 X Vy are eigenvectors of Pw and state their corresponding eigenvalues. (d) Find a diagonal matrix D and an...
Need help with linear algebra problem! Let S be a symmetric, 2 x 2 matrix. Let û1) and ût2) be orthogonal eigenvectors of S with corresponding nonzero eigenvalues A1 and X2. Show that if v E R2 is a vector such that û1)Su = 0, then 5 = Bû(2) for some B 0. Let S be a symmetric, 2 x 2 matrix. Let û1) and ût2) be orthogonal eigenvectors of S with corresponding nonzero eigenvalues A1 and X2. Show that...
Please show full workings only answer if you know how. (5) Consider the 3 x 3 matrix A - I - avv7 where a e R. I is the identity matrix and v the vector 1S 2 (a) Determine the eigenvalues and eigenvectors of A (b) Hence find a matrix which diagonalises A. (c) For which a is the matrix A singular? (d) For which α is the matrix A orthogonal ? (5) Consider the 3 x 3 matrix A...
(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P such that PT AP is a diagonal matrix Hint : UseLO and Problem EK〗 (g) Let A be a square matrix and Rn × Rn → Rn is defined by: UCTION E AND MES FOR THE la(x, y) = хтАУ (i) Show that I is symmetric, ie, 14(x,y) = 1a(y, x), if a d Only if. A is symmetric (ii) Show that...
3. Find all the eigenvalues and corresponding eigenspaces for the matrix B = 4. Show that the matrix B = 0 1 is not diagonalizable. 0 4] Lo 5. Let 2, and 1, be two distinct eigenvalues of a matrix A (2, # 12). Assume V1, V2 are eigenvectors of A corresponding to 11 and 22 respectively. Prove that V1, V2 are linearly independent.
4. Consider 3 linearly independent vectors V1, V2, V3 E R3 and 3 arbi- trary numbers dı, d2, d3 € R. (i) Show that there is a matrix A E M3(R), and only one, with eigenvalues dı, d2, d3 and corresponding eigenvectors V1, V2, V3. (ii) Show that if {V1, V2, V3} is an orthonormal set of vectors. then the matrix A is symmetric.
(5) Consider the 3 x 3 matrix A = 1-ovyT where the vector E R, 1 is the identity matrix and v (a) Determine the eigenvalues and eigenvectors of A. b) Hence find a matrix which diagonalises A. c) For which a is the matrix A singular? (d) For which a is the matrix A orthogonal ? (5) Consider the 3 x 3 matrix A = 1-ovyT where the vector E R, 1 is the identity matrix and v (a)...
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2 + 2 ) 2 16. + Problem 24. Show that: (a+b+c+d) (- [5 marks] Problem 25. Given any TEC (V) on an inner product space V define: [u, u] = (T(u),T(0) Is (u, v) (u, v) an inner product? If not, then provide conditions on T such that this becomes an inner product, and prove this completely. (5 marks Problem 26. Suppose TEC(V) and dim range T = k. Prove that I has at most k + 1 distinct...