Eigenvectors If A is × n. recall that an eigenvector is a vector u € R\" such th
ID: 3117481 • Letter: E
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Eigenvectors If A is × n. recall that an eigenvector is a vector u € R" such that Ar =Awfr scene scalar · The scalar is called the eigenvalue corresponding to the eaptvertor r. set of all eigenvectors with the same eigenvalue is a vector space called an eigemspace To find eigenvalues, we solve the polynomial equation det (Al-A) = 0·The polynomial p(A) = det(Al - A) is called the characteristic polynomial of the matrix A Given an n xn matrix A, it will have n-eigenvalues (alt hough they may not all be distinct). Let Dbe the n × n diagonal matrix whose only entres are exactly the n-einvalues of A (placed along the dia paul). let P be the n × n matrix whose i-th column is a basis of the eigetspoe corresponding to the eigenvalue in D. Then A PDP -1 17· Let A= 0 3-1 (a) Find the eigenvalues of A (e) Find a diagonal matrix D and an invertible matrix P such that A-PDP 18. Let A 1 2 1 0-13 For all eigenvalues, find a basis for the corresponding eigenspace Find the eigenvalues of A For all eigenvalues, find a basis for the corresponding eigenspace (b) (c) Find a diagonal matrix D and an invertible matrix P such that A- PDP 19·LetA- 010 (a) Find the eigenvalues of A (c) Find a diagonaal matrix D and an invertible matrix P such that A = PDP-I 1 7 0 13-2 For all eigenvalues, find a basis for the corresponding eigenspace 1-2 8 20· Let A- 0-1 0 Find the eigenvalues of A (b) For all eigenvalues, find a basis for the corresponding eigenspace (e) Find a diagonal matrix D and an invertible matrix P such that A PDP Linear Transformation A function f : Rn Rm is linear is f(ar +bur) = af(r) +bf(w) wherea and bare umbers and e and te are vectors in R Recall that every such function f can be represented by a matrix A with the function (r) given by matrix maltiplication Ae where e is represented as a colmn vector To contract the mstrix A, simply let the ith dolunanof A be the m × 1 (okamn) vector y(q) where is the ith standard basis vector of Rn. el (1,0 ,0), e2-(R 1, 0, ,0)es (0,0,,0),n(0,.,0,1) etc. (a) Find that matrix that represents f (b) Compute f(-1, 4) using matrix multiplicationExplanation / Answer
17.(a). The eigenvalues of A are solutions to its characteristic equation det(A-I3) = 0 or, 3-102+32-32 = 0 or,(-4)2(-2) = 0. Hence the eigenvalues of A are 1 =2= 4 and 3 =2.
(b).The eigenvectors of A associated with its eigenvalue 4 are solutions to the equation (A-4I3)X = 0. The RREF of A-4I3 is
0
1
1
0
0
0
0
0
0
Thus, if X = (x,y,z)T, then the equation (A-4I3)X = 0 is equivalent to y+z = 0 or, y = -z. Then X = (x,-z,z)T = x(1,0,0)T+z(0,-1,1)T. Hence, the eigenvectors of A associated with its eigenvalue 4 are v1 =(0,-1,1)T and v2 =(1,0,0)T. Similarly, the eigenvector of A associated with its eigenvalue 2 is (1,1,1)T.
For, the eigenvalue 4, the eigenbasis of A is E4 = { (1,0,0)T,(0,-1,1)T}.
For, the eigenvalue 2, the eigenbasis of A is E2 = { (1,1,1)T}.
(c ).The columns of P are v1, v2 and v3 and the entries on the main diagonal of A are 1,2 and 3.
Hence P =
1
0
1
0
-1
1
0
1
1
and D =
4
0
0
0
4
0
0
0
2
21.(a). We have f(x,y) = (-x+y,y). Hence f(1,0) = (-1,0) and f(0,1) = (1,1). Therefore, the standard matrix of f is A =
-1
1
0
1
(b). f(-1,4) = A(-1,4)T = (5,4)T
Pleasae pose the remaing questions again, one at a time.
0
1
1
0
0
0
0
0
0
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