Show that the column rank and the row rank of A are both equal to 3.

Transcription

1 hapter Vectors and matrices.. Exercises. Let A Show that the column rank and the row rank of A are both equal to Let x and y be column vectors of size n, andleti be the identity matrix of size n. (i) If y t x, show that (I + xy t ) I +y t x xyt. In other words, show that I +y t x xyt (I + xy t ) I. (ii) Show that the matrix I + xy t is nonsingular if and only if y t x. 3. (i) Use induction to show that! t A i i i A t n+ i for any m i n i matrices A i, i :n, withn i m i+ for i :(n ). (ii) Show that! A i i i A n+ i for any nonsingular square matrices A i of the same size.

2 2 HAPTER. VETORS AND MATRIES 4. Let D diag(d i) i:n be a diagonal matrix of size n with distinct diagonal entries, i.e., such that d d k, for any k n. IfA is a square matrix of size n, show that AD DA if and only if the matrix A is diagonal. 5. Use the fact that D D 2 D 2D for any two diagonal matrices D and D 2 of thesamesizetoshowthat D i D p(i), i for any one-to-one function p : {, 2,...n} {, 2,...n}, whered i, i :n, are diagonal matrices of the same size. 6. (i) Let A be an n n matrix and let L be an n n nonsingular lower triangular matrix. Show that, if LA is a lower triangular matrix, then A is lower triangular. Show that, if AL is a lower triangular matrix, then A is lower triangular. (ii) Let A be an n n matrix and let U be an n n nonsingular upper triangular matrix. Show that, if UA is an upper triangular matrix, then A is upper triangular. Show that, if AU is an upper triangular matrix, then A is upper triangular. 7. Let A be a nonsingular matrix, and let k be a positive integer. Define A k as the k th power of the inverse matrix of A, i.e., let A k `A k. Show that this definition is consistent, i.e., show that i A k A k A k A k I. 8. (i) Let ompute M 2, M 3, M 4. M 3 2 (ii) Let I + M 3 2 ompute m,wherem 2 is a positive integer. Hint: Recall that, if A and are square matrices of the same size such that A A, then the following version of the binomial formula holds true: mx (A + ) m m A m, (.)

3 .. EXERISES 3 m where m is a positive integer and the binomial coefficient is given by m m!! (m )!, where k! 2... k. Also, note that A I. 9. Let L be an n n lower triangular matrix with entries equal to on the main diagonal, i.e., with L(i, i) for i : n. (i) Show that L n ; (ii) ompute (I + L) m in terms of L, L 2,...,L n,wherem n is a positive integer. Hint: Use the binomial formula (.).. Let A and be square matrices of the same size with nonnegative entries and such that the sum of the entries in each row is equal to. Show that the matrix A has the same properties, i.e., show that all the entries of the matrix A are nonnegative and the sum of the entries in each row of A is equal to. Note: A matrix with nonnegative entries such that the sum of the entries in each row is equal to is called a probability matrix.. The covariance matrix of five random variables is Σ Find the correlation matrix of these random variables. 2. The correlation matrix of five random variables is Ω A (i) ompute the covariance matrix of these random variables if their standard deviations are.25,.5,, 2, and 4, in this order. (ii) ompute the covariance matrix of these random variables if their standard deviations are 4, 2,,.5, and.25, in this order.

4 4 HAPTER. VETORS AND MATRIES 3. The file indeces-ul26-aug9-22.xlsx from fepress.org/nla-primer contains the July 26, 22 August 9, 22 end of day values of Dow Jones, Nasdaq, and S&P 5. (i) ompute the daily percentage returns of the three indices over the given time period. (ii) ompute the covariance matrix of the daily percentage returns of the three indices. (iii) ompute the daily log returns of the three indices over the given time period. (iv) ompute the covariance matrix of the daily log returns of the three indices. Note: The percentage return and the log return between times t and t 2 of an asset with price S(t) attimet are given by S(t 2) S(t ) S(t2) and ln, S(t ) S(t ) respectively. 4. The file indices-uly2.xlsx from fepress.org/nla-primer contains the January 2 July 2 end of day values of nine maor US indices. (i) ompute the sample covariance matrix of the daily percentage returns of the indices, and the corresponding sample correlation matrix. ompute the sample covariance and correlation matrices for daily log returns, and compare them with the corresponding matrices for daily percentage returns. (ii) ompute the sample covariance matrix of the weekly percentage returns of the indices, and the corresponding sample correlation matrix. ompute the sample covariance and correlation matrices for weekly log returns, and compare them with the corresponding matrices for weekly percentage returns. (iii) ompute the sample covariance matrix of the monthly percentage returns of the indices, and the corresponding sample correlation matrix. ompute the sample covariance and correlation matrices for monthly log returns, and compare them with the corresponding matrices for monthly percentage returns. (iv) omment on the differences between the sample covariance and correlation matrices for daily, weekly, and monthly returns. 5. In three months, the value of an asset with spot price $5 will be either $6 or $45. The continuously compounded risk free rate is 6%. onsider the one period market model with two securities, i.e., cash and the asset, and two states, i.e., asset value equal to $6 and asset value equal to $45, in three months.

5 .. EXERISES 5 (i) Find the payoff matrix of this model. (ii) Is this one period market complete, i.e., is the payoff matrix nonsingular? (iii) How do you replicate a three months at the money put option on this asset, using the cash and the underlying asset? 6. In six months, the price of an asset with spot price $4 will be either $3, $35, $4, $42, $45, or $5. onsider a one period market model with six states in six months corresponding to the six possible values of the asset in six months, and with the following four securities: cash; asset; six months at-the-money call option with strike $4 on the asset; six months at-the-money put option with strike $4 on the asset. The continuously compounded risk free interest rate is constant and equal to 6%. (i) Find the payoff matrix of this model. (ii) Is this one period market model complete? (iii) Are the four securities non redundant?