1. ## Matrix Linear Equations

Anyone able to help me verify?

Consider a sequence of real-number r1, r2, ....,rN
Using this we create an (N * N) square matrix R such that the (i, j)th element of R is given by rK , where k = min(i,j), i = 1, 2, ...., N ; j = 1, 2, .....,N

A. Write the matrix R in terms of its elements. Clearly, show at least the top 3 × 3 part
and all the elements on the four corners.

r11 r1 2 r13 ------ r1N * x1 = b1
r21 r22 r23----- -r2N * x2 = b2
r31 r32 r33--------r3N * x3 = b3
ri1 ri2 ri3-------riN * xj = bj
rm1 rm2 rm3------ rmn * xN = bN

B. Is this a symmetric matrix?
yes?
A = A^T

Regards & a million thanks
Chris

2. Originally Posted by Chris0724

Consider a sequence of real-number r1, r2, ....,rN
Using this we create an (N * N) square matrix R such that the (i, j)th element of R is given by rK , where k = min(i,j), i = 1, 2, ...., N ; j = 1, 2, .....,N

A. Write the matrix R in terms of its elements. Clearly, show at least the top 3 × 3 part
and all the elements on the four corners.

B. Is this a symmetric matrix?
the matrix R is obviously symmetric because $\displaystyle \min(i,j)=\min(j,i).$ here's the matrix: $\displaystyle R=\begin{pmatrix} r_1 & r_1 & r_1 & . & . & . & r_1 \\ r_1 & r_2 & r_2 & . & . & . & r_2 \\ r_1 & r_2 & r_3 & . & . & . & r_3 \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ r_1 & r_2 & r_3 & . & . & . & r_N \end{pmatrix}$ . can you show that $\displaystyle \det R=r_1 \prod_{j=1}^{N-1} (r_{j+1}-r_j)$ ?

3. thanks for the help

i can't seem to understand the meaning where k = min(i,j), i = 1, 2, ....N; j = 1, 2,....N.

Able to explain? thanks a million

4. Hello,

"matrix R such that the (i, j)th element of R is given by $\displaystyle r_k$ , where $\displaystyle k=\min(i, j)$, i = 1, 2, ...., N ; j = 1, 2, .....,N" simply means that "the (i, j)th element of R is $\displaystyle r_{\min(i, j)}$ (i=1,...,N;j=1,...,N)."
For example, the (2, 3)th element of R(usually denoted $\displaystyle R_{23}$) is $\displaystyle r_{\min(2, 3)}$. Since 2 is less than 3, $\displaystyle \min(2, 3)=2$. Thus, the (2, 3)th element of R is $\displaystyle r_2$.

Bye