Problem 2 Show that a square matrix A can be written as the sum of a symmetric a

Question

Problem 2 Show that a square matrix A can be written as the sum of a symmetric and skew-symmetric matrices. (Hint: define B A+AT and C-A-A) Problem 3: Markov Process Suppose that the bus ridership in a city is studied. After examining several years of data, it was found that 35% of the people who regularly ride on buses in a given year do not regularly ride the bus in the next year. Also it was found that 25% of the people who do not regularly ride the bus in that year, begin to ride the bus regularly the next year. Knowing that 5500 people ride the bus and 15,000 do not ride the bus in a given year, Determine how many people will ride the bus next year. Write the system of equations in equivalent matrix form Ax-b Determine the distribution of riders/non-riders in the next year? In 2 years? In n years? -

Explanation / Answer

Solution to problem 2

Let, A be any square matrix.

Now, A=0.5(A+A)+0.5(AA)=P+Q

where P = 0.5(A+A) and Q = 0.5(AA)

To prove: P is symmetric , i.e. P = P and Q is skewsymmetric, i.e. Q = Q

P = 0.5(A+A) = 0.5(A+(A)) = 0.5(A+A) = P

P is symmetric

Q = 0.5(AA) = 0.5(A(A)) = 0.5(AA) = Q

Q is skewsymmetric

To prove uniqueness, let A = R + S where R is symmetric and S is skewsymmetric

A = (R+S) = R+S = RS

{ R = R and S = S by Definition of symmetric and skewsymmetric matrices }

0.5 (A+A) = 0.5(R+S+RS) = R = P

0.5(AA)= 0.5(R+SR+S) = S = Q

Hence, the representation A = P+Q is unique. Hence, it is proved that every square matrix can be uniquely expressed as a sum of symmetric and skew-symmetric matrix.

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