# 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 .

281 questions
Filter by
Sorted by
Tagged with
31k views

### $AB-BA=I$ having no solutions

The following question is from Artin's Algebra. If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions. I have no idea on how to tackle this question. I ...
• 6,489
8k views

### Solutions to the matrix equation $\mathbf{AB-BA=I}$ over general fields

Some days ago, I was thinking on a problem, which states that $$AB-BA=I$$ does not have a solution in $M_{n\times n}(\mathbb R)$ and $M_{n\times n}(\mathbb C)$. (Here $M_{n\times n}(\mathbb F)$ ...
• 2,309
6k views

### 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 ...
• 407
2k views

### Prove that matrix can be square of matrix with real entries

Prove that matrix \begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix} can be square of matrix with all real entries. I have found one such matrix to be \begin{bmatrix}1&...
• 3,993
3k views

### For every matrix $A\in M_{2}( \mathbb{C})$ there's $X\in M_{2}( \mathbb{C})$ such that $X^2=A$?

True\False? For every matrix $A\in M_{2}( \mathbb{C})$ there's $X\in M_{2}( \mathbb{C})$ such that $X^2=A$. I know that every complexed matrix has a Jordan form matrix $J$ such that $P^{-1}CP=J$, ...
7k views

### $AB=BA$ implies $AB^T=B^TA$ when $A$ is normal

I am looking for an elementary proof (if such exists) of the following: $$AB=BA \quad\Longrightarrow\quad AB^T=B^TA,$$ where $A$ and $B$ are $n\times n$ real matrices, and $A$ is a normal matrix, i....
• 84.2k
5k views

### Is there a unique solution for this quadratic matrix equation?

Here is the quadratic matrix equation I've been looking at lately: $$Q_{r,r}=A_{r,r}X_{r,r}^2+B_{r,r}X_{r,r}+C_{r,r}=0_{r,r}$$ Note that $A, B, C,$ and $X$ are $r \times r$ matrices. $A$ contains ...
• 5,760
173 views

### Matrix equation with symmetric and positive definite matrices [duplicate]

Let matrices $P_0$ and $P_1$ be symmetric and positive definite. Is it possible to find a symmetric matrix $S$ such that $SP_0S=P_1$? If $P_0=I$, this is always possible: $S=\sqrt{P_1}$.
• 405
11k views

### 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 $A^{-1}$ ...
• 802
1k views

### 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 $n\times n$ ...
• 2,817
5k views

• 166
3k views

### Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$ [duplicate]

I have the following question : Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$ I managed to proof that $I+BA$ invertible My proof : We know that $AB$ and $BA$ ...
• 2,726
3k views

### Tangent Space of SL(n,R) at arbitrary point, e.g. not at $\mathbb{1}$

I am looking for the tangent space of $SL(n,\mathbb{R}) = \{A\in\mathbb{R}^{n\times n} | \det{A}=1 \}$ (actually $n=3$) at an arbitrary $M\in SL(n,\mathbb{R})$. From now on I will omit the $\mathbb{R}$...
• 607
### If matrix $AB=A$, does it mean B must be an identity matrix?
If matrix multiplication $AB=A$, does it mean $B$ must be an identity matrix? If not, why? What conditions? $A$ is not a zero matrix.