Are there any solutions to $Ax=b$ satisfying $\Vert x \rVert=1$?

Question

Let $$A = \begin{pmatrix} 1 & 1 & -3 \\ -3 & -2 & -2 \\ -7 & -5 & 1 \end{pmatrix}$$ be a $3$ by $3$ matrix and $$b = \begin{pmatrix} 4 \\ 6 \\ 8 \end{pmatrix}$$ be a column matrix. Are there any solutions to $Ax = b$ satisfying $\Vert x \rVert = 1$? Please explain, I am a bit confused on how to approach this, I know it's connected to the min norm solutions.

Any help will be appreciated.

Answer 1

This is a problem with norms. You need to know that the norm of a matrix $A$ is defined as the maximum norm you get for a vector $Ax=b$ when $x$ is a vector with norm $1$.

Let us use the $L_1$ vector norm which is the sum of the absolute values of the components, meaning in this case $4+6+8=18$ for your target product vector.

Now the corresponding $L_1$ norm for the matrix $A$ is the maximum of the norms of its column vectors. Here the first column has $L_1$ norm equal to $11$, the second column gives $8$ and the third column $6$. So the overall $L_1$ norm of your matrix is only $11$, thus the norm of the product $Ax$ with $x$ having unit norm can't reach the value of $18$ for your given $b$ vector. Hence our answer based on this norm is No.

Answer 2

The following solution requires a computational toolbox like to Wolfram Alpha invert a matrix The inverse of A is given by $A^{-1} = \begin{pmatrix} -6 & 7 & -4 \\ 9.5 & -10 & 5.5 \\ 0.5 & -1 & 0.5 \end{pmatrix}$. Multiply it by $b$ and you get the solution $x = \begin{pmatrix} -14 \\ 18 \\ 0 \end{pmatrix}$.

Answer 3

To solve your problem you must find the solution of $Ax=b$. You can use any technique you are familiar with to solve this system. You will arrive that the solution is the vector

$$x = \begin{pmatrix} -14 \\ 18 \\ 0 \end{pmatrix}$$

And the norm of the solution is $\lVert x \rVert = \sqrt{520} =2\sqrt{130}$.

Note: In this case I am using the usual norm, that is $\lVert x \rVert =\sqrt{(-14)^2+18^2+0^2}$.

As suggested by Bruno in his answer, it is possible to solve this system by finding the inverse matrix of $A$, but another technique that is also widely used is the Gaussian elimination method. Sugestion [1] and [2].