Consider the eigenvalue decomposition of a symmetric matrix a. prove that two eigenvectors vị and v; associated with two distinct eigenvalues li and l; of a are mutually orthogonal; that is, v. vj = 0

Lets consider the symmetric matrix , and the two eigenvectors and such as:

with

.

The dot product between and can be obtained with:

Using the first eigenvector equation we can find:

Lets transpose it

as is an scalar

Now, as A is symmetric:

so

Lets take the dot product again:

but this is :

But, the parenthesis is equal to

so

Now, subtracting the dot product

As the eigenvalues are distinct,   can't be 1, so

this implies

so the eigenvectors are orthogonal.

rick has $1400 and sally has $1500

step-by-step explanation:

to find amount of money rick has after 10 years :

principal = $1000 , time = 10 years , rate = 4% annually

to find amount of money sally has after 10 years :

her friend gives to pay her $50 per year, therefore money she has after 10 years = 50 × 10 = $500

also, sally gets $1000 after 10 years,

therefore, total amount of money sally has after 10 years = 500 + 1000 = $1500

so, after 10 years : rick has $1400 and sally has $1500

i only know the sequence of arithmetic it adds the first is 5+4 then it goes up as it adds to then it is 9+5 then 14+6 then to 20+7 and so on. 5,9,14,20,27,35.

step-by-step explanation: