Questions tagged [matrix-equations]

This tag is for questions related to equations, with matrices as coefficients and unknowns. A matrix equation is an equation in which a variable stands for a matrix .

Definition: Let $~v_1,~v_2,~\cdots~,v_n~ $ and $~b~$ be vectors in $~\mathbb{R^n}~$. Consider the vector equation $$x_1~v_1+x_2~v_2+~\cdots~+x_n~v_n=b~$$This is equivalent to the matrix equation$$~Ax=b~$$

where $~~A=\begin{pmatrix} \cdot & \cdot & \cdots & \cdot \\ v_1 & v_2 & \cdots & v_n \\ \cdot & \cdot & \cdots & \cdot \\ \end{pmatrix};~~ x=\begin{pmatrix} x_1 \\ x_2\\ \cdots\\ x_n \end{pmatrix} ~~\text{and}~~ b=\begin{pmatrix} b_1 \\ b_2\\ \cdots\\ b_n \end{pmatrix}$

Since a matrix equation $ ~AX=B~$ (where $ ~X~$ is a column vector of variables) is equivalent to a system of linear equations, we can use the same methods we have used on systems of linear equations to solve matrix equations. Namely:

$(1.)~~$ Write down the augmented matrix $ ~A \vdots B$.

$(2.)~~$ Row-reduce to a new augmented matrix $~ \overline A \vdots \overline B~$ in row echelon form.

$(3.)~~$ Use this new matrix to write a matrix equation equivalent to the original one.

$(4.)~~$ Use this new, equivalent matrix equation to find the solutions to the original equation.

In mathematics, matrix equation (which is a system of linear equations) is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system.

Very often, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring. Integer linear programming is a collection of methods for finding the "best" integer solution (when there are many). Gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure.

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Find the inverse of a submatrix of a given matrix

I have a $A$ matrix $4 \times 4$. By delete the first row and first column of $A$, we have a matrix $B$ with sizes $3 \times 3$. Assume that I have the result of invertible A that denote $A^{-1}$ before. I want to use the result to calculate the…
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What is the inverse of the $\mbox{vec}$ operator?

There is a well known vectorization operator $\mbox{vec}$ in matrix analysis. I've vectorized my matrix equations, did some transformation of vectorized equations and now I want to get back to the matrix form. Is there special operator for it?
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Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ matrices with real entries. Let…
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Matrix equation in characteristic 2

Let $n\geq 3$ be an odd integer and $K$ be a field with characteristic $2$. Let $A,B\in M_n(K)$ s.t. $A^2+B^2=I_n$; is it true that $AB+BA$ is a singular matrix? Remark. i) It is not difficult to see that the result is false when $n$ is even. ii)…
user91684
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Find $A^{1000}$ by using Cayley-Hamilton Theorem

I get stuck at the following question: Consider the matrix $$A=\begin{bmatrix} 0 & 2 & 0 \\ 1 & 1 & -1 \\ -1 & 1 & 1\\ \end{bmatrix}$$ Find $A^{1000}$ by using the Cayley-Hamilton theorem. I find the characteristic polynomial by $P(A) = -A^{3} +…
surfer1311
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what do free variable and leading variables mean?

What do the leading variables and free variables in a matrix mean? I have the system below and am trying to understand which are which. I searched a lot for this, please help me ! $$w + x + y + z = 6 \qquad w + y + z = 4 \qquad w + y = 2$$
user136980
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Find matrix $A\in \mathcal{M}_n (\mathbb{N})$ such that $A^k =\left( \sum_{i=1}^{k}10^{i-1} \right)A$.

I was watching this video by Flammable Maths about why $$ \begin{pmatrix} 3 &4\\ 6&8 \end{pmatrix}^2 = \begin{pmatrix} 33 &44\\ 66&88 \end{pmatrix} $$ In the video, it is left as a challenge for the viewer to see if you can generalize the result as…
Robert Lee
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Does this $3\times 3$ matrix exist?

Does a real $3\times 3$ matrix $A$ that satisfies conditions $\operatorname{tr}(A)=0$ and $A^2+A^T=I$ ($I$ is an identity matrix) exist? Thank you for your help.
user74200
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Finding the 10th root of a matrix

I want to find a $2 \times 2$ matrix, named $A$ in this situation, such that: $$A^{10}=\begin {bmatrix} 1 & 1 \\ 0 & 1 \end {bmatrix} $$ How can I get started? I was thinking about filling $A$ with arbitrary values $a, b, c, d$ and then multiplying…
fxcd
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How should I study The Matrix Cookbook?

I use The Matrix Cookbook by Kaare Brandt Petersen and Michael Syskind Pedersen to solve many problems (mostly matrix derivatives). In most cases, I just map the problem to one of the formula and solve it but I cannot derive the formula by myself…
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Is this possible? AB- BA=I

I have just started linear functionals when I faced the following problem: If $A$ and $B$ are $n \times n$ complex matrices, show $AB - BA=\Bbb{I}$ is impossible. Can someone help me?
Qwerty
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If $A^2=2A$, then $A$ is diagonalizable.

I think, I should use a double linear transformation but can't find any proper solution. Let $\mathbb F$ be a field, $\mathscr M_n (\mathbb F)$, the set of $n\times n$ matrices with elements in $\mathbb F$, and $A\in \mathscr M_n (\mathbb F)$…
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On the complex matrix equation $AX-XA=B$

I want to show that there exists solution to the matrix equation $AX-XA=B$ if and only if $$ \begin{pmatrix} A&0\\ 0&A \end{pmatrix}, \begin{pmatrix} A&B\\ 0&A \end{pmatrix} $$ are similar, where all matrices ($A,B,X$) are complex and $A,B$ are…
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Matrix version of Pythagoras theorem

Can I find a solution for $C_{n\times n}$, explicitly, for the given $A_{n\times n}$ and $B_{n\times n}$ such that $AA^{T} + BB^{T} = CC^{T}$? Here $A^{T}$ denotes the transpose of $A$ and all the matrices ($A$, $B$, and $C$) are real. I think…
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Number of solutions of a matrix algebraic equation

Reading that, I found that: given a matrix algebraic equation $$ X^n+A_1X^{n-1}+\cdots + A_n=0 $$ where the cofficients $A_1\cdots,A_n$ as well as solutions $X$ are supposed to be square complex matrices of some order $k$....... generically…
Emilio Novati
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