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Suppose I have a matrix $A$. I want to add a scalar to its diagonal or all elements in order to make its determinant equal to one. Is this possible?

For example:

$\textbf{B} = \textbf{A} + n$ where $\det(\textbf{B})=1$

$\textbf{B} = \textbf{A} + n\textbf{I}$ where $\det(\textbf{B})=1$ and $\textbf{I}$ is an identity matrix

$\textbf{B} = \textbf{A} + n\textbf{I} + m$ where $\det(\textbf{B})=1$ and $\textbf{I}$ is an identity matrix

$m$ and $n$ are scalar and the will be added to the all element of matrices. I want to identify $n$ or $m$ (or both) in order to make $\det(\textbf{B})$ equal to $1$.

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If $A$ is an $n\times n$ matrix where $n$ is odd then the answer is yes, and you only have to add $\lambda I$:

$$\det(A + \lambda I) = (-1)^n\det(-A - \lambda I) = -k_{-A}(\lambda)$$

Since $-k_{-A}(\lambda)$ is a polynomial of degree $n$, it is surjective, so for some $\lambda$ it will be $-k_{-A}(\lambda) = 1$.

Edit:

Let $A = (a_{ij})$ be an $n\times n$ matrix. The characteristic polynomial $k_A$ of the matrix $A$ is defined as:

$$k_A(x) := \det(A - xI) = \begin{vmatrix} a_{11} - x & a_{12} & \ldots & a_{1n}\\ a_{21} & a_{22}-x & \ldots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} - x\\ \end{vmatrix}, \quad x\in\mathbb{R}$$

Notice that $k_A$ is indeed a polynomial in $x$ because the determinant is defined as: $\DeclareMathOperator{\sgn}{sgn}$

$$\det A = \sum_{p \in S_n} (\sgn p) \cdot a_{1p(1)}a_{2p(2)}\cdots a_{np(n)}$$

that is, a sum of all products of $n$ elements of $A$ such that from each row and column of $A$ exactly one element was taken.

Also notice that $k_A$ is a polynomial of degree $n$ as the power of $x^n$ is obtained through the product of the elements on the main diagonal: $(a_{11} - x)(a_{22} - x)\cdots (a_{nn}-x)$.

If you wish to find an $x \in \mathbb{R}$ such that $\det(A - xI) = 1$, you are actually solving the equation $k_A(x) - 1 = 0$.

Now, if $n$ is odd then $k_A$ is a polynomial of an odd degree, which is surjective (because when $x\to\pm\infty$ we have $k_A(x)\to\pm\infty$, and $k_A$ attains all points in between).

This means that you can certainly find such $x$, you just have to find the zeros of the polynomial:

$$a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + (a_0 - 1) = 0$$

where $k_A(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$.

In general, you won't be able to do this by hand, so you can for example use Wolfram Mathematica.

If $n$ is even, then $k_A$ is not surjective, so it could happen that there doesn't exist a value $x$ such that $k_A(x) = 1$.

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  • $\begingroup$ I am not mathematician. Could you please explain what is k and its subscribe A. How can calculate lambda? $\endgroup$ – user1436187 Sep 14 '17 at 2:07
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    $\begingroup$ @user1436187 I have added an explanation. $\endgroup$ – mechanodroid Sep 14 '17 at 11:09

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