Question: The trace of a n Ã n matrix A MnÃn. is defined to be the sum of its diagonal entries: tr A = a11 +

The trace of a n Ã— n matrix A ˆˆ MnÃ—n. is defined to be the sum of its diagonal entries: tr A =
a11 + a22 +ˆ™ ˆ™ ˆ™ + ann.
(a) Compute the trace of

(b) Prove that tr(A + B) = tr A + tr B.
(c) Prove that tr(A B) = tr(BA).
(d) Prove that the commutator matrix C = A B - B A has zero trace: tr C = 0.
(e) Is pan (c) valid if A has size m Ã— n and B has size n Ã— m?

211 303 114 3 2

Step by Step Solution

★★★★★

3.46 Rating (162 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a i 4 ii 0 b c The diagonal entries of AB are t... View full answer

Question Has Been Solved by an Expert!

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

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

952-M-L-A-E (1610).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

If A = [ajj] is an n Ã n matrix, then the trace of A, Tr(A), is defined as the sum of all elements on the main diagonal of A, Show each of the following: (a) Tr(cA) = c Tr(A), where c is a...

Let U be an m m matrix, V be an n n matrix, and let where 1 is an n n diagonal matrix with diagonal entries 1, 2,..., n and O is the (m - n) n zero matrix. (a) If U = (U1, U2), where U1 has n...

The trace of an n n matrix A Mnn is defined to be the sum of its diagonal entries: tr A = a11 +a22 H+. . . + ann. Prove that the set of trace zero matrices, tr A = 0, is a subspace of Mmn?

Trace of a Matrix Using the following definition, prove the properties of the trace in Problems 1-3. The trace of an n x n matrix A = [aij], denoted Tr A, is the sum of the diagonal elements: 1....

The trace of an n X n matrix A = [aij] is the sum of the entries on its main diagonal and is denoted by tr(A). That is, tr(A) = a11 + a22 + + ann If A and B are n n matrices, prove the following...

........... Write the system of linear equations in the form Ax = b and solve this matrix equation for x. X1 - 5X2 + 2X3 = -13 -3X1 + X2 - X3 = 10 -2X2 + 5X3 = -17 X1 - 13 X2 10 X3 -17 X1 X2...

Prove Theorem 4.2. [A proof by induction would be appropriate here.] In theorem 4.2 The determinant of a triangular matrix is the product of the entries on its main diagonal. Specifically, if A =...

MATHEMATICS FOR MACHINE LEARNING Marc Peter Deisenroth A. Aldo Faisal Cheng Soon Ong Contents Foreword 1 Part I Mathematical Foundations 9 1 Introduction and Motivation 11 1.1 Finding Words for...

chapter 5 INTRODUCTION TO MATRIX ALGEBRA GOALS The purpose of this chapter is to introduce you to matrix algebra, which has many applications. You are already familiar with several algebras:...

If Alexei has 4 times as many quarters as dimes and they have a combined value of 440 cents, how many of each coin does he have?

A bird flies 15 km [S 25 E] and then 20 km [S 30 W] to reach its destination. How would this look on a diagram?

26. 1. Open the general ledger accounts and accounts payable ledger accounts indicated below. Enter the balances as of June 1, 20X1. 2. Post the transactions in Problem 8.4A to the appropriate...

KEY QUESTION Assume that a hypothetical economy with an MPC of .8 is experiencing severe recession. By how much would government spending have to increase to shift the aggregate demand curve...

Is there a matrix analogue of formula (1.56), namely A T B = B T A? vw = (vw) = wv, W

List the diagonal entries of A = 97 2.6 13 37 48 5 9 10 11 14 15 8 12 16,

Determine which of the following are bases of R 3 : (a) (b) (c) (d) 1 (-3), ( -2 5

The Snedecker Corporation is considering a change in its cash-only policy. The new terms would be net one period. The required return is 1.6 percent per period. Based on the following information,...

QUESTION 6 If the simple CAPM is valid and all portfolios are priced correctly, which of the situations below is possible? Consider each situation independently, and assume the risk-free rate is 5%....

what is a compensating control in security controls?