北美Math数学代写-MTH 342代写-Linear Algebra代写

MTH 342 OSU Spring 2022

北美Math数学代写 Before you compute A1/2 in part (e), explain why this is going to give us a square root of A. In other words, explain the equality

Introduction.

On Worksheet 2, you computed f(A) for the polynomial f(x) =x 2 − 2x − 2 and the matrix

It is possible to compute g(A) for other functions g that are not polynomials. For ex-ample, it is possible to compute g(A) where g is the exponential function g(x) = e x .Diagonalization of (diagonalizable) matrices is part of computing functions of matri-ces. In this worksheet we will see how diagonalization can be used to fifind the square roots of a matrix A. (A square root of matrix A is a matrix B such that B2 = A.)

1.This problem includes a review of how to fifind eigenvalues and eigenvectors of a 2 × 2 matrix. 北美Math数学代写

The purpose of this problem is to give you a guide you can consult when you do Problem 2.

Work through the following example of diagonalizing a 2 × 2 matrix (from LADWp.112). Consider the matrix

(a) Find its characteristic polynomial and check that it has roots λ = 5 and λ = −3.

(b) Check that (1, 2)T is an eigenvector corresponding to λ = 5. You can fifind this eigenvector (or a scalar multiple of it) by solving (A − 5I)~v = ~ 0.

(c) Check that (1, −2)T is an eigenvector corresponding to λ = −3. You can fifind this eigenvector (or a scalar multiple of it) by solving (A + 3I)~v = ~ 0.

(d) Check that the matrix A can be diagonalized as

(Do the matrix multiplication.) Observe that the columns of S are the eigen-vectors above.

2. Let A be the following matrix:

In this problem you will diagonalize A to fifind its square roots. A square root of matrix C is a matrix B such that B2 = C. A given matrix C can have multiple square roots.

(a) Start by diagonalizing A as A = SDS−1 (see Problem 1).

(b) Then compute one of the square roots D1/2 of D. The square-roots of a diagonal matrix are easy to fifind.

(d) Let A1/2 = SD1/2S −1 . Before you compute A1/2 in part (e), explain why this is going to give us a square root of A. In other words, explain the equality

A¹^/²A¹^/² = A.

(e) Compute A¹^/². This is just one of several square root of A (you only need to compute one of them, not all of them.) Your fifinal answer should be a 2 × 2 matrix with all of the entries computed.

(f) How many distinct square roots does A have?

3. Suppose A is an n×n matrix such that A = SDS−1 where D = diag{d1, d2, . . . , dn} is a diagonal matrix, and S is an invertible matrix. 北美Math数学代写

Prove that the columns of S are eigenvectors of A with corresponding eigenvalues being the diagonal entries of D.

Hint: let the columns of S be 1, 2, . . . ,n. Compute AS in two difffferent ways and

then compare the results:

(i) Compute it as AS = A[1 2 · · · n]. Use (∗) below.

(ii) Compute it as AS = (SDS−1 )S = SD.

(∗) Suppose A and B are matrices with matrix product AB. If b1, b2, . . . , br are the columns of B, then Ab1, Ab2, . . . , Abr are the columns of AB. (p. 19 LADW)