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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2
votes
1answer
143 views

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)...
2
votes
1answer
107 views

Find a $4\times 4$ matrix whose reduced row echelon form has two leading ones and whose row space intersects its column space only at the origin.

Find a $4\times 4$ matrix whose reduced row echelon form has two leading ones and whose row space intersects its column space only at the origin. Would $$ \begin{bmatrix}1& 0& 1&0\\0&...
2
votes
2answers
42 views

Proof of $W=M_{n}(R)$

Let $n$ be an integer $\ge 2$ and let $M_{n}(R)$ denotes the vector spaces of $n\times n$ real matrices. Let, $B\in M_{n}(R)$ be an orthogonal matrix & let $B^{T}$ be the transpose of $B$. ...
2
votes
2answers
41 views

How to show that $\| QA\|_2=\| A \|_2$ where $Q$ is unitary (for a matrix A)

I want to show that for a unitary matrix $Q$ and a matrix $A$ that $$ \|QA\|_2=\|A\|_2$$ I start with the definition of matrix induced norms: $$\| QA \|_2 = \sup_{x \neq 0}\frac{\|QAx\|_2}{\|x\|_2}...
2
votes
1answer
58 views

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 ...
2
votes
2answers
52 views

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?
2
votes
2answers
310 views

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 ...
2
votes
2answers
50 views

Preservation of rank implies Invertibility

Show that if the rank of $XY$ (where $Y$ is an $n\times n$ matrix) is the same as the rank of $X$ for every $m\times n$ matrix $X$, then $Y$ is invertible. I thought I had found a counterexample: $$ ...
2
votes
3answers
154 views

How to find a symmetric matrix $B$ where $x^T Ax = x^T Bx.$ [duplicate]

Find a symmetric matrix $B$ such that for every $3\times 1$ matrix $x$. $$x^T Ax = x^T Bx\ .$$ Let $$A = \begin{pmatrix}2& 1& -1\\3& 0& 1\\-2& 5& 3\end{pmatrix}$$
2
votes
3answers
103 views

Getting back the column vector from which a matrix was generated

We know that given a vector $\mathbf{X}\in \mathrm{R}^{n\times 1}$, we can create a matrix $\mathbf{A} = \mathbf{XX}^T$, where $\mathbf{A}\in\mathrm{R}^{n\times n}$. Now let us suppose the reverse ...
2
votes
2answers
52 views

$A \in Gl(n,K)$ if and only if $A$ is a product of elementary matrices.

I want to show the statement: $A \in GL(n,K)$ if and only if $A$ is a product of elementary matrices. I could show the reverse implication: Suppose $A=T_1...T_m$ where each $T_k$ is an elementary ...
2
votes
1answer
44 views

Find inverse of $I+\mathbf{ab}^\intercal$

Could you guys give me some hints on this homework? Find inverse of $\mathbf{I} + \mathbf{ab}^\intercal$. Hint: try to form $c\mathbf{I} + d\mathbf{ab}^\intercal$ and solve for $c,d$. What happens ...
2
votes
2answers
341 views

How to find eigenvalues of the following block circulant matrix

I have a block matrix of size PN x PN of the form: Where A and C are P x P matrices. I would like to find the eigenvalues of the matrix B, that is where
2
votes
2answers
4k views

How to find a basis of an image of a linear transformation?

I apologize for asking a question though there are pretty much questions on math.stackexchange with the same title, but the answers on them are still not clear for me. I have this linear operator: $$...
2
votes
2answers
62 views

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 ...
2
votes
2answers
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, ...
2
votes
2answers
542 views

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 ...
2
votes
3answers
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 ...
2
votes
2answers
74 views

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 $A$ ...
2
votes
2answers
279 views

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 ...
2
votes
2answers
44 views

A result on extension fields in linear algebra.

Let $F$ be a subfield of $E$, $A$ an element of $\mathcal{M}_F(m,n)$ and $b$ a vector in $F^m \subset E^m$. What is the easiest way to prove the following statement: if $Ax = b$ has a solution in $...
2
votes
3answers
7k views

find all values of k for which A is invertible

$\begin{bmatrix} k &k &0 \\ k^2 &2 &k \\ 0& k & k \end{bmatrix}$ what I did is find the det first: $$\det= k(2k-k^2)-k(k^3-0)-0(k^3 -0)=2k^2-k^3-k^4$$ when $det = 0$ ...
2
votes
1answer
54 views

is some of matrice with it's transpose positive definite? when eigenvalues of matrix is positive

Suppose M = A+ A^T , and we know that all of eigenvalues of A are real and positive, is M positive definite? or semi positive definite?
2
votes
1answer
99 views

Matrix Groups in Abstract Algebra

QUESTION: Let $h= \begin{pmatrix} -1 & 1\\-1&0 \end{pmatrix} \in GL_2(\Bbb R)$. Find $\langle h\rangle$. I'm stuck on the solution, but here is what I have: Let $h=\begin{pmatrix} -1&1\\-...
2
votes
2answers
243 views

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 ...
2
votes
2answers
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.
2
votes
1answer
124 views

Find $e^{AT}$ where $A$ is a Matrix that is given

How to find the value of $e^{At}$ where $A$ is the matrix $A =\begin{bmatrix} 4 & 3 \\ 2 & -1 \end{bmatrix}$
2
votes
2answers
6k views

Explicit formula for inverse of upper triangular Toeplitz matrix inverse

I have $n \times n$ upper triangular matrix $A$ such as $$ \begin{bmatrix} x_1 & x_2 & \ldots & x_n \\ 0 & x_1 & \ldots & x_{n-1} \\ \vdots & \vdots & \...
2
votes
1answer
165 views

Prove that matrices have equal rank.

If $P$ and $Q$ are $n \times n$ matrices of real numbers such that $P^2=P$ and $Q^2=Q$ and $I-P-Q$ is invertible where $I$ is an $n \times n$ identity matrix, Show that $P$ and $Q$ have the ...
2
votes
2answers
496 views

Normal but not hermitian nor unitary

I have to find out a normal transformation that is neither hermitian nor unitary. http://en.wikipedia.org/wiki/Normal_matrix gives me the answer. However, I would like to know how to find it out ...
2
votes
2answers
82 views

A non-nilpotent matrix $A\in \mathbb C^{2 \times2}$ has a square root

Is there any quick argument to show that every non-nilpotent matrix $A\in \mathbb C^{2 \times2}$ has a square root? Just the existence without computing it. Knowing that $A\in \mathbb C^{2 \times2}$ ...
2
votes
1answer
1k views

If a set of 2x2 matrices are independent, do they also span M22?

I am looking for a basis in M22. So I need to get a linearly independent set that also spans the vector space. I have worked out how to tell independence, but I am stuck on the spanning requirement. ...
2
votes
3answers
99 views

Explain why $S$ is not a basis for $\mathbb{R}^3$

Explain why $S$ is not a basis for $\mathbb{R}^3$ $S=\{(1, 3, 0),(4, 1, 2),(-2, 5, -2)\}$ I set this equal to an arbitrary vector $\mathbf{x} = (x_1, x_2, x_3)$ After solving I got the matrix: $\...
2
votes
1answer
211 views

Graph of a matrix

How to define the graph of a square matrix $\mathbf{G}$ with real entries? I know that given a graph $\Gamma(V, E)$, one can define its adjacency matrix $\mathbf{A}$. But given a matrix $\mathbf{G}$ ...
2
votes
1answer
46 views

How do you handle this kind of probability?

What is the probability of selecting a singular matrix from $\Bbb{R}^{3\times 3}$? I calculated it to be zero based on their being approximately $9$ degrees of freedom to choose entries of $A$ such ...
2
votes
1answer
63 views

Jordan similar matrix

I have matrix $B = \begin{bmatrix}1 & 1 & -2 & 0\\2 & 1 & 0 & 2 \\ 1 & 0 & 1 & 1 \\ 0 & -1 & 2 & 1\end{bmatrix}$. I found the characteristic polynomial $...
2
votes
1answer
104 views

Conditions for “$AA^T=A^TA$ implies $A$ symmetric” to hold.

This claim arose in this question Show that $A$ is symmetric, with $A \in M_n(\mathbb R)$ where it is assumed additionally that $AA^TA$ is symmetric. I'm considering weakening hypotheses. Let $A$ be ...
2
votes
3answers
385 views

Show that $M_2(\mathbb{R})$ has no non-trivial two-sided ideals

In addition to the title question, I also want to find a non-trivial right ideal and a non-trivial left ideal of $M_2(\mathbb{R})$ . Attempt of title question: Suppose $\exists I\subset M_2(\mathbb{...
2
votes
2answers
58 views

Find amount of invertible matrices of size $3 \times 3$ over residue field modulo 5

Find amount of invertible matrices of size $3 \times 3$ over residue field modulo 5. I will just add that this task is slightly ahead of my knowledge of field theory. So any pointers would be ...
2
votes
2answers
70 views

Spectral Radius and Norm of multiplied vector

Let $\mathbf{A}$, $\mathbf{B}$ be square matrices of equal dimensions, $\mathbf{w}$ a vector of compatible dimensions and $\rho$ be the spectral radius operator. Does the following hold? If $\rho (...
2
votes
1answer
2k views

Find high powers of a matrix with the Cayley Hamilton theorem

Let A = \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ -1 & -1 &-1\\ \end{bmatrix} Compute $A^{10000} + A^{9998}$ I know this should be done by the Cayley-Hamilton theorem. I get ...
2
votes
2answers
76 views

Matrix-Vector Product

Suppose I have the expression $\lVert\mathbf B \cdot\hat n\lVert=1$, where $\mathbf B$ is a matrix and $\hat n$ is a unit vector (both can have any dimensions, as long as they are compatible). What ...
2
votes
2answers
63 views

the spectrum of a matrix

If $A$ is an $n \times n$ nilpotent matrix show that $I-A$ is invertible then find the spectrum of I-A ? for part one i've shown that $I-A$ is invertible by finding its inverse using that $A$ is a ...
2
votes
1answer
940 views

Matrix Exponential using the Cayley-Hamilton theorem

For the matrix $$P=\left( \begin{matrix} 0&1&0 \\ 0&0&1 \\ 0&1&0 \end{matrix} \right)$$ how do you find $e^{Pt}$ using the Cayley-Hamilton theorem? I have found it by ...
2
votes
1answer
3k views

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 ...
2
votes
2answers
53 views

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 ...
2
votes
1answer
456 views

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 ...
2
votes
2answers
2k views

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} &...
2
votes
2answers
109 views

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 ...