Is the matrix a full rank matrix?

Question

Assume $\phi_1(x),\phi_2(x),...\phi_n(x)$ are linear independent, $$A = \begin{pmatrix} \phi_1(x_1) & \phi_2(x_1) & \cdots & \phi_n(x_1) \\ \phi_1(x_2) & \phi_2(x_2) & \cdots & \phi_n(x_2) \\ \vdots & \vdots& \vdots & \vdots \\ \phi_1(x_m) & \phi_2(x_m) & \cdots & \phi_n(x_m) \end{pmatrix}$$

Is $A$ a full rank matrix? ($m>n$),$x_1,x_2,...x_m$ are different.

I believe that m>n implies this matrix will either have linearly dependent columns or rows and is therefore not full-rank but is rather either full row rank or full column rank. — 123, Dec 27 '14 at 02:32
Now I have a matrix A like this,In the software 'MATLAB',I calculate the rank of A'*A, it says the rank is not equal to n,So I guess the A is not necessarily full rank. — yang, Dec 27 '14 at 02:37

score 1 · Answer 1 · answered Dec 27 '14 at 05:54

We know that $sin (x) $ and $cos(x)$ are linearly independent. Let's take $n=2$ and $m=3$. Then consider the points (${\pi \over 4}, {\pi\over 4}+2\pi, {\pi\over 4}+4\pi$). Both $sin(x)$ and $cos(x)$ take the same values at these three points.So, the columns of the matrix are linearly dependent.

user1537366 · Answer 2 · 2014-12-27T03:32:06.050

Suppose $\phi_k$ are polynomials of degree $<m$.

For all linear combinations $(c_k)$ of the columns:

Let $q=\sum_k c_k\phi_k$. $$\sum_k c_k\left(\array{\phi_k(x_1)\\\phi_k(x_2)\\\vdots\\\phi_k(x_m)}\right)=0\implies \left(\array{q(x_1)\\q(x_2)\\\vdots\\q(x_m)}\right)=0\implies q=0\implies(c_k)=(0)$$

So the columns are linearly independent.

Otherwise, let $(a_{k,j})_{n\times n}$ be an invertible matrix, and $M$ bigger than all $x_j$.

Let $\phi_k$ be curves (even polynomials of degree $m$) each passing through all the points $(x_j,0)\forall j=1,\ldots,m+1-n$ and also $(M+j,a_{k,j})\forall j=1,\ldots,n$. It is easy to see that they are "linearly independent", but the matrix will have at least $m+1-n$ zero rows, and have a rank of at most $m-(m+1-n)=n-1$, so the columns are linearly dependent.

But the $\phi_k$ are not necessarily polynomials,for example, $\phi_k=\sqrt{1+(ε_k(x−c_k)^2}$, $c_k$ are distinct. — yang, Dec 27 '14 at 03:04

score 0 · Answer 3 · edited Apr 13 '17 at 12:20

While it'd be true for polynomials, for example, a counterexample with $sin$, $\cos$ is given in @Srinivas's answer. Fortunately, your functions are positive definite as I already pointed out to you. Positive definite functions have their associated interpolation matrices positive definite. Hence the $n\times n$ matrix would be invertible and $m\times n$ of full rank. You can read more about this class of functions in the same textbook I referred you to last time.

Is the matrix a full rank matrix?

3 Answers3