Question: Let V = Mn(R), A be a fixed nn matrix, and T: V V be the linear map defined by T(B) = AB. i.

Let V = Mn(R), A be a fixed n×n matrix, and T: V −→ V be the linear map defined by T(B) = AB. 

i. Show that the minimal polynomial for T is the minimal polynomial for A. 

ii. Show that if A is diagonalizable over R, then T is diagonalizable.

Step by Step Solution

★★★★★

3.39 Rating (149 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer
Step: 1 Unlock

i The minimal polynomial of A mAt is a monic polynomial of degree n such that mA... View full answer

blur-text-image
Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock
Step: 3 Unlock

Students Have Also Explored These Related Accounting Questions!

Q:

Explain why the linear map defined by - I defines a rotation in two-dimensional space, but a reflection in three-dimensional space.

Q:

Let A be a diagonalizable matrix with characteristic polynomial (a) If D is a diagonal matrix whose diagonal entries are the eigenvalues of A, show that = p(D) + a1Dn + a2Dn-1 + an+1I = 0 (b) Show...

Q:

Let h be the linear map defined by this matrix on the domain P1 and codomain R2 with respect to the given bases. What is the image under h of the vector = 2x - 1? 1 2 2 1 B (1 + x,x), D

Q:

Let x a (t) be a bandlimited signal with fixed bandwidth B and variance 2 x . (a) Show that the signal-to-quantization noise ratio. SQNR = 10log 10 ( 2 x / 2 x ). Increase by 3 dB each time we...

Q:

Kalton and Anderson (1986) consider disproportional stratified random sampling for estimating the mean of a characteristic yi in a rare population. Let ri = 1 if person i is in the rare population...

Q:

Kalton and Anderson (1986) consider disproportional stratified random sampling for estimating the mean of a characteristic yi in a rare population. Let ri = 1 if person i is in the rare population...

Q:

Let U be a fixed n n matrix, and consider the operator T: Mnu M"" given by T(A) = UA. (a) Show that A is an eigenvalue of T if and only if it is an eigenvalue of U. (b) If A is an eigenvalue of T,...

Q:

The annulus shown below is cut from a planar metal sheet with thickness t and conductivity . (a) Let V be the voltage between the edge CD and the edge FA. Solve Laplaces equation to find the...

Q:

Kalton and Anderson (1986) consider disproportional stratified random sampling for estimating the mean of a characteristic yi in a rare population. Let ri = 1 if person i is in the rare population...

Q:

Kalton and Anderson (1986) consider disproportional stratified random sampling for estimating the mean of a characteristic yi in a rare population. Let ri = 1 if person i is in the rare population...

Q:

Exercise 8-26 (LO. 8) Parent and Child corporations have filed on a consolidated basis since the mid-1970s. The group reports the following amounts for the current tax year. Operating loss, including...

Q:

1) BobTheCat Enterprises has asked FoxFirst Consulting to help with upgrading 600 desktop users from Windows 7 to Windows 10. As per contract, FoxFirst will provide the upgrade strategy. Discuss...

Q:

Harvey's Restaurants In your own words, explain the Harvey's brand, Canadian values, and range of products/services. A minimum of four sentences is required here. Strengths Explain three (3)...

Q:

Find the initial basic feasible solution for the TP by VAM. D D2 D3 Dy Supply Dz 0, 11 13 17 14 250 02 16 16 18 14 10 300 03 21 24 13 10 400 Dem 200 225 275 250 950

Q:

3. Both Hawks Nest and the Upper Big Branch mine were underground work locations. Does this have any significance in the hiding of safety violations? If so, how? 4. Why would a modern organization...

Q:

Any device which carries data between end devices? The answer i picked is Intermediary Device is that correct if not what would the answer be?

Q:

Jamesway Corporation makes two types of replacement fittings for heavy construction equipmentscrews and bolts. Data regarding the two products follow: Direct Labour-Hours per Unit Annual Production...

Q:

Draw two scatterplots, one for which r = 1 and a second for which r = 21.

Q:

(a) Give an example of a non-diagonal 22 matrix for which A2 = I. (b) If A2 = I, show that det A = 1

Q:

Construct a symmetric matrix with the following eigenvectors and eigenvalues, or explain why none exists: (a) 1 = 1, v1 = (3/5, 4/5)T, 2 = 3, v2 = (-4/5, 3/5)T (b) 1 = -2, v1 = (1, -l)T, 2 = l, v2 =...

Q:

Prove that if we apply a unit force to node i in a structure and measure the displacement of node j in the direction of the force, then we obtain the same value if we apply the force to node j and...

Q:

28.Astatistician is interested in testing the hypotheses and H0:p> 120 H^.ft

Q:

16. A hot dog vendor at a football game must decide in advance how many hot dogs to order. He makes a profit of $.10 on each hot dog which is sold, and he suffers a $.20 loss on hot dogs which are...

Q:

20. In Exercises 15, 16, 17, and 19, what would the optimal estimates be if the loss function were quadratic instead of linear [that is, if the loss function were of the form (9.13.1)]?