Find the determinant of the following matrix

Question

Find the determinant of the following matrix: $$A = \begin{bmatrix} 1+x_1^2 &x_1x_2 & ... & x_1x_n \\ x_2x_1&1+x_2^2 &... & x_2x_n\\ ...& ... & ... &... \\ x_nx_1& x_nx_2 &... & 1+x_n^2 \end{bmatrix}$$

I computed for the case $n=2$, and $n=3$ and guessed that $\det(A)$ should be $ 1+\sum_{i=1}^n x_i^2 $ but not sure how to proceed for any $n$.

The matrix is $I_n + XX^T$, where $X = \left(x_1, x_2, ..., x_n\right)^T$. The matrix $XX^T$ has rank $\leq 1$. So you are looking for the determinant of a "rank-$1$ update" of a matrix, or for the characteristic polynomial of the rank-$\leq 1$ matrix $XX^T$. Either interpretation suggests a way of computing it (I think the latter is better known). — darij grinberg, Jan 12 '15 at 19:28
A more general case was treated here: Determinant of rank-one perturbation of a diagonal matrix — hardmath, Jan 14 '15 at 02:31

Ewan Delanoy · Answer 1 · 2019-03-05T05:34:37.720

8

To expand on darij grinberg's comment, let

$$ X=A-I_n = \begin{bmatrix} x_1^2 &x_1x_2 & ... & x_1x_n \\ x_2x_1&x_2^2 &... & x_2x_n\\ ...& ... & ... &... \\ x_nx_1& x_nx_2 &... & x_n^2 \end{bmatrix}=(x_ix_j)_{1\leq i,j\leq n} $$

Then all the lines of $X$ are multiples of $(x_1,x_2,\ldots,x_n)$ ; so ${\textsf{rank}}(X)\leq 1$. The eigenvalues of $X$ (counted with multiplicity) are therefore $0,0,\ldots,0$ ($n-1$ times), plus some $\lambda\in{\mathbb R}$. Since the trace of $X$ equals the sum of its eigenvalues, we must have $\lambda={\textsf{trace}}(X)=\sum_{i=1}^n x_i^2$. Then $X$ is similar to a triangular matrix with diagonal ${\textsf{diag}}(0,0,0,\ldots, 0,\sum_{i=1}^n x_i^2)$, so that $A$ is similar to a triangular matrix with diagonal ${\textsf{diag}}(1,1,1,\ldots, 1,1+\sum_{i=1}^n x_i^2)$, whence

$$ {\textsf{det}}(A)=1+\sum_{i=1}^n x_i^2 $$

edited Mar 05 '19 at 05:34

answered Jan 12 '15 at 20:18

Ewan Delanoy

63,436

On the third line below the matrix, do you mean $\lambda = trace(X)$? Why is that so? – Paul555 Jan 13 '15 at 16:14
@Paul555 see my update – Ewan Delanoy Jan 13 '15 at 16:15
1

Note: Instead of saying "$X$ is similar to ${\textsf{diag}}(0,0,0,\ldots, 0,\sum_{i=1}^n x_i^2)$", it may be better to say "$X$ is similar to a triangular matrix with diagonal $(0,0,0,\ldots, 0,\sum_{i=1}^n x_i^2)$", since this frees us from having to prove that $X$ is diagonalizable. – darij grinberg Mar 05 '19 at 04:12
@darijgrinberg Indeed. Corrected, thanks – Ewan Delanoy Mar 05 '19 at 05:35

Davide Giraudo · Answer 2 · 2015-01-23T16:52:28.993

Here is a method without using (at least explicitely) the notion of eigenvalue.

Call $f(x_1,\dots,x_n)$ the wanted determinant. View the last column as $$\pmatrix{0\\\vdots\\ 0\\1 }+x_n\pmatrix{x_1\\\vdots\\ x_{n-1} \\x_n }.$$ This gives by linearity with respect to the last column, $$f(x_1,\dots,x_n)=f(x_1,\dots,x_{n-1})+x_n\det\begin{bmatrix} 1+x_1^2 &x_1x_2 & ...&x_1x_{n-1} & x_1 \\ x_2x_1&1+x_2^2 &... &x_2x_{n-1}& x_2\\ ...& ... & ... &&... \\ x_nx_1& x_nx_2 &...& x_nx_{n-1} & x_n \end{bmatrix}$$ (the first determinant is computed by expanding with respect to the last column). In the last determinant, do $C_i\leftarrow C_i-x_iC_n$, $1\leqslant i\leqslant n-1$ in order to show that $$\det\begin{bmatrix} 1+x_1^2 &x_1x_2 & ...&x_1x_{n-1} & x_1 \\ x_2x_1&1+x_2^2 &... &x_2x_{n-1}& x_2\\ ...& ... & ... &&... \\ x_nx_1& x_nx_2 &...& x_nx_{n-1} & x_n \end{bmatrix}=x_n.$$ Now we can conclude by induction.

Mark · Answer 3 · 2015-01-12T21:24:46.140

2

Consider the eigenvalues of $x \cdot x^T + I$

$x \cdot x^T v + v = \lambda v$
$x \cdot x^T v = \lambda v - v$

$v$ must be parallel to $x$ or ($\lambda$ = 1). wlog $v = x$

$||x|| ^2 x= (\lambda - 1) x$
$\lambda = ||x||^2 + 1$

You can use the fact that the determinant is the product of eigenvalues (there should be $n$ of them)

edited Jan 12 '15 at 21:24

answered Jan 12 '15 at 19:38

Mark

6,038

how do we know that there is only one nonzero $\lambda$ and all other eigenvalues are 0 ? – Paul555 Jan 12 '15 at 23:13
The other eigenvalues are all 1. If $\lambda = 1$ we see that the equation is satisfied iff $v$ perpendicular to $x$. This eigenspace is $n-1$ dimensional, leaving only the final dimension for the non-1 eigenvalue. – Mark Jan 13 '15 at 04:53
Thanks, Mark. Could you also explain why $v$ must be parallel to $x$ ? – Paul555 Jan 13 '15 at 16:33
You see that the range of $x \cdot x^T$ lies entirely in the line of $x$. This is because whenever you multiply by a vector $v$ on the right, $x^T v$ turns into a real number and you get out something parallel to $x$. – Mark Jan 13 '15 at 17:19

Find the determinant of the following matrix

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