3. (a) (4 marks) Let A be an n x n matrix with entries in R. Define what it mean

Question

3. (a) (4 marks) Let A be an n x n matrix with entries in R. Define what it means for v to be an eigenvector of A and define what it means for 2 to be an eigenvalue of A. (b) For the matrix 4=[[ ]] =3[13] (i) (8 marks) Find the eigenvalues and an orthonormal eigenbasis or show that no such eigen- basis exists. (ii) (4 marks) Provide a diagonalisation of A. (iii) (4 marks) Sketch a picture that indicates geometrically the action of the transformation Ta : R2 ? R2. (c) (5 marks) Explain what it means for a matrix in Mn(R) to have complex eigenvalues. For the L 1 1 matrix A = , describe geometrically the linear transformation Ta : R2 ? R2. | -1 1 Total: 25 marks

Explanation / Answer

3.a). If A is a nxn matrix, then a scalar ? is called an eigenvalue of A if there is a non-trivial solution X of the equation AX = ?X. Further, such a vector X is called an eigenvector corresponding to the eigenvalue ?.

b). (i)The eigenvalues of A are solutions to its characteristic equation det(A-?I2) = 0 or, ?2-3?+2 = 0 or,   (?-2)(?-1)= 0. Thus, ?1= 2 and ?2 =1 are the 2 eigenvalues of A. Further, the eigenvector of A corresponding to the eigenvalue 2 is solution to the equation (A-2I2)X= 0. To solve this equation, we will reduce A-2I2 to its RREF which is

1

-1

0

0

Now, if X = (x,y)T, then the equation (A-2I2)X= 0 is equivalent to x -y = 0 or, x = y. Then, X = (y, y)T = y(1,1)T. Hence the eigenvector of A corresponding to the eigenvalue 2 is v1 = (1,1)T. Similarly, v2 = (1,-1)T is the eigenvector of A corresponding to the eigenvalue 1. The eigenvectors of A are orthogonal as v1.v2 = 0. Now, let u1 = v1/||v1|| = (1/?2, 1/?2)T and u2 = v2/||v2|| = (1/?2, -1/?2)T. Then {u1,u2} is the required orthonormal eigenbasis.

(ii). Let P = [v1,v2] =

1

1

1

-1

And D = diag[?1, ?2]=

2

0

0

1

Then A = PDP-1

Please post the other parts separately again.

1

-1

0

0

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