A square matrix P is called an idempotent if P2 = P. Show that: If P is

Question:

A square matrix P is called an idempotent if P2 = P. Show that:
If P is an idempotent, so is
Q = P + AP - PAP for any square matrix A (of the same size as P).

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Linear Algebra With Applications

ISBN: 9780070985100

7th Edition

Authors: W. Keith Nicholson

See More Books

Question Posted: Jun 21, 2016 06:26 AM

See More Questions

An n Ã n matrix P is called an idempotent if P2 = P. Show that: (a) I is the only invertible idempotent. (b) P is an idempotent if and only if I - 2P is self-inverse. (c) U is self-inverse if...
Let V be a vector space. If f: V [R is a linear transformation and z is a vector in V, define Tj z: V V by Tf.z(v) = f(v)z for all v in V. Assume that f 0 and z 0. (a) Show that Tj z is a linear...
A projection matrix is a square matrix P that satisfies P2 = P. (a) Prove that w mg P if and only if P w = w. (b) Show that mg P and ker P are complementary subspaces, as defined in Exercise 2.2.24,...
Asset allocation explains a large portion of a portfolio return. However, the implementation issues involved inthe asset allocation process may reduce the efficiency of the asset allocation strategy,...
This problem requires the use of ACL software, which is included in the CD attached to the text. Information about installing and using ACL and solving this problem can be found in Appendix, pages...
In negotiations for the purchase of a ranch, Adler (the seller) discussed water rights with Folt, the buyer. Adler never mentioned an ongoing dispute she had over such rights with a neighboring...
The treatment assignment displayed in Table 2.5 is from one treatment assignment out of $\left(\begin{array}{l}20 \\ 10\end{array} ight)$ possible random treatment assignments. a. $\mathrm{R}$ can be...
Required Identify which of the following items are added to or subtracted from the unadjusted book balance to arrive at the true cash balance. Distinguish the additions from the subtractions by...
More generally, evaluate the new Federal Government policies: 1 What are the pros and the cons of the FDIC guarantee of debt? 2. Pros and cons of executive compensation restrictions? 3. Pros and cons...
Richard chooses technique 0 and 2 requiring 10+10-20 efforts and provising 10+11=21 benefits. Hence, 21 is returned as the output Example 2: input1: 3 input2: (10,10,10,10) input3: (10,11,12,15)...
The trace of a square matrix A, denoted tr A, is the sum of the elements on the main diagonal of A. Show that, if A and B are n n matrices: (a) tr(kA) = k tr(A) for any number k. (b) tr(AAT) is the...
If A and B are n n matrices, show that: (a) AB = BA if and only if (A + B)2 = A2 + 2AB + B2. (b) AB = BA if and only if (A + B)(A - B) = (A - B)(A + B).
Find the dimensions of a rectangle with a perimeter of 148 feet that has the maximum area.
1. Solve the following problems involving combined operations. (7+9) x 6 = b. 2 X (22+34) + 8 = 25 x (8+5)-16= d. (154-72) x 43+ (108-96)= e. 53+ 25+ (154 - 39) = f. 5,496-553+ (6 x 55) 72 = 2. A...
To answer the following questions, perform an Internet search for sample proposals. For example, type project proposal example into Google or Yahoo. Based on the results of your search, find a sample...
The Farmer's Market just paid an annual dividend of $5 on its stock. The growth rate in dividends is expected to be a constant 5 percent per year indefinitely. Investors require a 13 percent return...
Assume in analyzing alternative proposals that Proposal F has a useful life of six years and Proposal J has a useful life of nine years. What is one widely used method that makes the proposals...
A car is driving toward you with a speed of 86 km/hr. The roof racks on the car emit a whistling sound, which you hear as a 514 Hz tone. If you are stationary, what will the frequency of this tone be...
What is the going-concern assumption? What is a going-concern problem? What factors can contribute to such a problem?
d) For die casting processes: 1. What are the most common metals processed using die casting and discuss why other metals are not commonly die casted? 2. Which die casting machines usually have a...
Determine the matrix representation of each of the following composite transformations. (a) A yaw of 90, followed by a pitch of 90 (b) A pitch of 90, followed by a yaw of 90J (c) A pitch of 45,...
Let Y, P, and R be the yaw, pitch, and roll matrices given in equations (1), (2) and (3) and let Q = YPR. (a) Show that Y, P, and R all have determinants equal to 1. (b) The matrix Y represents a yaw...
Let L be a linear operator on Rn. Suppose that L(x) =0 for some x 0. Let A be the matrix representing L with respect to the standard basis [e1, e2,..., en]. Show that A is singular
The Power of Trade and Comparative Advantage: Work It Out 3 ? Here's another specialization and exchange problem. This problem is wholly made-up, ? so that you won't be able to use your intuition...
Government spending as a fiscal policy tool is used to: A) ?Decrease the national debt B) ?Directly stimulate economic activity by increasing demand C) ?Reduce inflation D) ?Lower interest rates
Automobile demand has fallen during a recent recession, and Ana has been laid off from her job on the assembly line. Rajiv recently lost his job as a waiter at a local restaurant. A recent increase...

A square matrix P is called an idempotent if P2 = P. Show that: If P is

Question:

Step by Step Answer:

We have Q P AP PAP so si...View the full answer

Linear Algebra With Applications

Students also viewed these Linear Algebra questions