# Tagged Questions

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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### Positive definiteness of the matrix $A+B$

Let, $A$ & $B$ are $n\times n$ positive definite matrices & $I$ be the $n\times n$ identity matrix. Then which of the followings are positive definite? (a) $A+B$ (b) $ABA$ (c) $A^{2}+I$ (d)...
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### about the power of a matrix

Assume that matrix $A$ contains only 0 or 1 elements. Could anyone give me some condition, under which the matrices $A^i$ (for $i=1,2,3,...,k$) still contains only 0 or 1 elements. For example, I ...
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### if matrix multiplication $B*A=C*A$, does it mean $B=C$?

If matrix multiplication $B*A=C*A$, does it mean $B=C$? If A is invertible, then I guess this should work. If not, then?
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### If $AB = I$, the identity matrix prove $\mathrm{rank}(B)$

Let $A$ be an $m \times n$ matrix and $B$ be an $n \times m$ matrix. Show that if $AB = I$, where $I$ is the identity matrix, then $\mathrm{rank}(B) = m$. I'm not exactly sure how to start this ...
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### A question about invertible matrices

A square matrix $A$ over the reals is said to be invertible in practice if there exists a matrix $B$ of the same size s. t. all the entries of $AB$ differ from the corresponding entries of the ...
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365 views

### Newly Developed With Details - Describing orthographic projection using simple 2D transformations

Thanks to Pedro for helping me further develop my question into something tangible. His (most recent) answer below clearly and formally outlines what I am asking. This is similar to this question, ...
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### If $Ax = b$ has more than one solution so does $Ax = 0$, where $A$ is $m\times n$ real matrix.

Problem: If $Ax = b$ has more than one solution so does $Ax = 0$, where $A$ is $m\times n$ real matrix. In the explanation part it is written that when $Ax = b$ is consistent the solution sets of ...
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470 views

### Minimal polynomial of diagonalizable matrix

It's a if and only if sentence (have to prove both directions) If a matrix $A$ (over $\mathbb{C}$) is diagonalizable then its minimal polynomial's roots are all of algebraic multiplicity 1. Any idea ...
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### Matrices whose product is identity but do not commute.

I'm supposed find two matrices $A$ and $B$ whose product $AB=I_2$, but $BA\neq I$. But I'm not sure if this is even possible since if $AB=I$, doesn't that mean that $B$ is the inverse matrix of $A$ ...
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### Minimal polynomial and diagonalization of a block diagonal matrix. [duplicate]

Let $A \in \mathbb C^{m\times m}$ and $B \in \mathbb C^{n\times n}$, and let $C=\begin{pmatrix} A & 0 \\ 0 & B\\ \end{pmatrix} \in \mathbb C^{(m+n)\times (m+n)}$. Calculate the minimal ...
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### If $AX=XA$ for all $X$, then $A = \alpha I$ for some $\alpha$

Let $A$ be a $2 \times 2$ real matrix such that $AX=XA$ for all $2 \times 2$ real matrices $X$. Show that $A= \alpha I$ for some $\alpha ∈R.$ I am absolutely stuck, i thought $X$ and $A$ are ...
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239 views

### Is the square root of a symmetric positive definite matrix also symmetric?

The inverse of a SPD matrix is also symmetric. But what about the square root? Intuitively, I would say yes. But I'm not sure about it.
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### Find $e^{AT}$ where $A$ is a Matrix that is given

How to find the value of $e^{At}$ where $A$ is the matrix $A =\begin{bmatrix} 4 & 3 \\ 2 & -1 \end{bmatrix}$
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### Shortest distance matrix given an adjacency matrix?

If I have an adjacency matrix, how can I find a matrix that has the shortest distance between each pair of nodes? (distance matrix, but the nodes are not in a euclidean space) I'm trying to implement ...
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### Proof that matrix $B^{-1}$ = matrix $A^{-1}$ with 2 columns swapped given that B = A with 2 rows swapped.

I'm trying to prove the following. Given that $A$ is a nonsingular $n \times n$ matrix, and $B$ is the nonsingular matrix obtained by interchanging rows $i$ and $j$ of $A$, where $i \neq j$, show ...
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### center of invertible matrices [duplicate]

find the center of the group of invertible 2 x 2 matrices with real entries. Attempt: By definition, the center of a group Z(G), is where all the elements are commutative. If G = { invertible 2 x 2 ...
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### Expected number of steps between states in a Markov Chain

Suppose I am given a state space $S=\{0,1,2,3\}$ with transition probability matrix \$\mathbf{P}= \begin{bmatrix} \frac{2}{3} & \frac{1}{3} & 0 & 0 \\[0.3em] \frac{2}{3} &...
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### What is the structure of matrix multiplication and minus?

Please note I have only little background im mathematics and I am working on formalizing theorems with theorem provers. This is very much a beginner question. Suppose I have matrices, where the ...