Successive matrix multiplication

• sara_87
In summary, a matrix multiplying a vector of healthy and ill people produces a percentage of ill people after n days.
sara_87
On a given day of a flu epidemic, a given percentage Y of the population is ill and (1-Y) is healthy. The probability of remaining healthy on the next day is a, and that of remaining sick is B. The question is, what percentage will be ill after a given number of days as a function of B,a and Y.

Write down the problem as a matrix multiplication of the vector of healthy and ill people. Define a MATLAB function that gives the percentage of ill people after n days for arbitrary B a,and Y.

any hints will be appreciated v much
thank you

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sara_87 said:
On a given day of a flu epidemic, a given percentage Y of the population is ill and (1-Y) is healthy. The probability of remaining healthy on the next day is a, and that of remaining sick is B. The question is, what percentage will be ill after a given number of days as a function of B,a and Y.

Write down the problem as a matrix multiplication of the vector of healthy and ill people. Define a MATLAB function that gives the percentage of ill people after n days for arbitrary B a,and Y.

any hints will be appreciated v much
thank you

# healthy people = # people staying healthy + # new healthy people

The odds of staying healthy are a, so
# people staying healthy = a * # healthy people

New healthy people are sick people who did not stay sick. The odds of staying sick are B, so the odds of NOT remaining sick, i.e. new healthy people is (1-B), so
# new healthy people = (1-B) * # sick people

So if Y is the number of healthy people and X is the number of sick people at a given time,
Y[n+1] = a*Y[n] + (1-B)*X[n]

Write down a similar equation for X[n+1]. Do you see how this can be written as a matrix-vector product?

would the equation for X[n+1] be:
X[n+1]=B*X[n]+(1-B)*X[n]

i understood everything you did but I'm struggling to write it as a matrix vector product.
would it be something like:
(X,Y)=(somthing with a)+(something with b)
?
thank you

You want to write

$$\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} X_n \\ Y_n \end{bmatrix} = \begin{bmatrix} X_{n+1} \\ Y_{n+1} \end{bmatrix}$$

Multiplying this out gives

$$\begin{array}{c} X_{n+1} \\ Y_{n+1} \end{array} = \begin{array}{cc} A_{11}X[n] + A_{12}Y[n] \\ A_{21}X[n] + A_{22}Y[n] \end{array}$$

so find the coefficients for the A matrix that make this match your equations.

oh ok so A(21) would be=(1-B)
and A(22) would be=a
is that right?
and how do i find A(11) and A(12) because I'm not so convinced about my equation for X[n+1]??

1. What is successive matrix multiplication?

Successive matrix multiplication is the process of multiplying multiple matrices together in a specific order. This involves taking the product of two matrices and multiplying it by another matrix, and so on until all matrices have been multiplied together.

2. Why is successive matrix multiplication important?

Successive matrix multiplication is important in many areas of mathematics, computer science, and engineering. It is used to solve systems of linear equations, transform geometric shapes, and perform operations in computer graphics and data analysis.

3. What is the difference between successive matrix multiplication and regular matrix multiplication?

The main difference between successive matrix multiplication and regular matrix multiplication is the order in which the matrices are multiplied. In regular matrix multiplication, the matrices are multiplied in a specific order, while in successive matrix multiplication, the matrices are multiplied in a sequential order.

4. How do you determine the order of matrices in successive matrix multiplication?

The order of matrices in successive matrix multiplication is determined by the dimensions of the matrices. The number of columns in the first matrix must be equal to the number of rows in the second matrix, and so on for each subsequent multiplication. The final product will have the same number of rows as the first matrix and the same number of columns as the last matrix.

5. What are some practical applications of successive matrix multiplication?

Successive matrix multiplication has many practical applications, such as in data compression, image processing, and cryptography. It is also used in solving systems of linear equations in engineering and economics, and in calculating the probabilities of multiple events in probability and statistics.

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