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Is there an explicit formula for the inverse of matrix ? $$A_{i,j}=a_ib_j$$

What is this kind of matrix called?

EDIT: It turns out that $\det A=0$ because all the columns are propertional to each other so $A$ has no inverse.

EDIT: This problem is related to another question approximate a function by linear combination of its asympototics with fractional argument.

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    $\begingroup$ You mean $A$ is the product of a column vector $a$ and a row vector $b$ (of the same length)? Such a matrix will not be invertible (except in the trivial case where $a$ and $b$ and therefore $A$ are $1\times 1$, with $a_i, b_j \ne 0$). $\endgroup$
    – M. Vinay
    Commented Jun 5, 2014 at 3:44
  • $\begingroup$ Indeed @ EDIT :P $\endgroup$
    – M. Vinay
    Commented Jun 5, 2014 at 3:47
  • $\begingroup$ @M.Vinay Thanks for the answer. I just used mathematica to invert a 3-by-3 $A$ matrix and found out the $\det A=0$. $\endgroup$
    – user152166
    Commented Jun 5, 2014 at 3:47
  • $\begingroup$ What you can invert explicitely is $\alpha I + \beta A$ for $\alpha\neq 0$. $\endgroup$
    – Fabian
    Commented Jun 5, 2014 at 4:39

2 Answers 2

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The matrix can be written as a standard product in vector algebra $\mathbf{a} \mathbf{b}^T$. It is called direct vector product, dyadic product, or rank-1 matrix (unless, of course, $a$ or $b$ are the zero vector). It is also a special case of a Kronecker product.

In case $\mathbf{a} = \mathbf{b}$ and $|\mathbf{a}|=1$ it is also a projection matrix (on a 1d subspace).

And, as already mentioned by others, this matrix is only invertible in the trivial case.

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I'm not sure that this kind of matrix is called anything, but it must be of the form $$ \left[ \begin{array}{c} a_1 \\ \vdots \\ a_n \end{array} \right] [b_1 \ \cdots \ b_n]. $$ For such a matrix, notice that any two rows $i$ and $j$ with $a_i,a_j \neq 0$ will be equal to $a_i[b_1 \ \cdots \ b_n]$ and $a_j[b_1 \ \cdots \ b_n]$, and hence multiples of each other. A matrix with one row equal to a multiple of another row cannot be invertible (the rows would not be linearly independent). On the other hand if $a_i = 0$ for some $i$ then there is a zero row in the matrix and again the matrix is not invertible.

Thus a matrix of this form can never be invertible.

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