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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0
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1answer
42 views

Linear recurrent sequences and matrices.

Let $k$ be a field, let $d$ be an integer greater than $1$, let $(v,x)\in k^d\times k^d$ and let $A\in k^{d\times d}$ be invertible. For all $n\in\mathbb{N}$, let define the following element of $k$:...
3
votes
2answers
298 views

To show a set of projectors summing to the identity implies mutually orthogonal projectors

The general setting is the study of positive operator measures in quantum mechanics http://en.wikipedia.org/wiki/Positive_operator-valued_measure instead of the projector operator measures. Going ...
1
vote
2answers
22 views

The trace functional and its scalar multiples [duplicate]

I am trying to solve the following problem: Show that the trace functional on $n \times n$ matrices is unique in the following sense. If $W$ is the space of $n \times n$ matrices over the field $F$ ...
0
votes
1answer
54 views

Matrix addition and eigen values/vectors

If I start with matrix A given by $A = \begin{bmatrix}a & b \\ c & d \end{bmatrix}$ and I express it as a sum $A = \begin{bmatrix} w & x \\ y & z ...
1
vote
3answers
118 views

Prove that there are not two matrices 2x2 such that: $AB-BA=I_2$

I tried this question by multiplying explicitly the matrices but I think I'm not getting anything, so I think, well let's suppose false so $C(AB-BA)=C$ and find a contradiction but also I'm not ...
1
vote
3answers
68 views

Finding complex solution to $X^2 = A$

Let $A=\begin{pmatrix}2&3\\4&-2\end{pmatrix}$. (i) Find an invertible matrix $P$ such that $P^{-1}AP$ is diagonal. (ii) Find $A^n$ (for positive integers $n$). (iii) Find four (...
1
vote
0answers
11 views

Hessian matrix of the mahalanobis distance wrt the Cholesky decomposition of a covariance matrix

I'm stuck with the following problem: I have to compute the second derivative (hessian matrix) of the mahalanobis distance $$ [x-\mu]^{T} \Sigma^{-1} [x-\mu] $$ wrt to the Cholesky decomposition of ...
1
vote
1answer
28 views

Remainder modulo matrices.

Is it possible to have a consistent definition of $A\bmod_{left} B$ that respects matrix multiplication from left where $A$ and $B$ are two square matrices with non-negative entries? Is there a ...
0
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2answers
48 views

How to describe range of a linear transformation?

I'm self studying Linear Algebra from Hoffman Kunze, and I've come upon this problem. With complex number $z=x+iy$, $$T(z)=\begin{pmatrix} x-7y & 5y \\ -10y & x+7y \\ \end{pmatrix}$$ is ...
1
vote
1answer
40 views

Lipschitz continuity of $\sqrt{A}$

Let $U \subset\mathbb{R}^n$ be an open set, $\mathbb{S}^n$ be the set of all $n\times n$ symmetric real matrices, $A:U\to \mathbb{S}^n$ be a uniformly Lipschitz continuous function. Suppose $\exists ...
0
votes
4answers
168 views

A four square matrix and $A^5$(a raised to the power of 5)$=0$. Then $A^4=?$

A four square matrix and $A^5$(a raised to the power of 5)$=0$. Then $A^4=$ $I$(identity matrix) $-I$ $0$ $A$ My attempt: You can use the characteristic equation $$ A^2-Tr(A)A+I_2\det{A}=O_2,$$ ...
0
votes
0answers
49 views

Find the eigenvalues the block matrix $M=\begin{bmatrix}A+2D & A \\ A & D \end{bmatrix}$

Let $A$ be any square matrix with eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n$ and $D$ is a diagonal matrix with entries $d_1,d_2,\cdots,d_n$, then how can one find the eigenvalues of the ...
0
votes
0answers
23 views

Relationship between geometric multiplicity, algebraic multiplicity and left and right eigenvectors of a matrix

The following statement is from the book Matrix Analysis by Horn and Johnson. An eigenvalue λ with geometric multiplicity 1 can have algebraic multiplicity 2 or more, but this can happen only if ...
0
votes
1answer
27 views

Regarding element-wise derivative of matrices

Let $X=(x_{ij})$ be a real n-by-n matrix where $x_{ij}$ are in a range such that $X$ is insvertible. Let $Y=X^{-1}=(y_{pq})$, and regard $y_{pq}=y_{pq} (x_{11},x_{12},\ldots,x_{nn})$ as a function of ...
2
votes
2answers
17 views

Dumb question: Orthogonal complement of kernel = Row space

I'm really confused when trying to prove the following: $\mathrm{kern}(A)^\perp = \{y \in \mathbb{R}^n \mid y = z^T A, \ z \in \mathbb{R}^n \}$ The $\supseteq$ direction is easy: Let $z^T A \in$ ...
3
votes
1answer
72 views

Is there a name for matrices that are symmetric along the cross diagonal? [duplicate]

Something like $$ A= \begin{bmatrix} a & b & c\\ b & d & e\\ c & e & f \end{bmatrix} $$ would be a symmetric matrix because the values are reflected along the ...
0
votes
1answer
74 views

Is this a matrix notation of standard error?

What is the matrix notation of the standard error? A friend is referencing the standard error as: $$SE^2=(XX^T)^{-1}\sigma^2$$ $$\sigma^2 = \frac{1}{n-p}\sum\limits_{i=1}^n |\hat{y_i}-y_i|$$ ...
10
votes
3answers
102 views

Can one factor matrices?

I know that one can factor integers as a product of prime numbers. Is there an analog of it to matrices? Can we define prime matrices such that every matrix is a product of prime matrices? Is there ...
17
votes
2answers
4k views

$AB-BA=I$ having no solutions

The 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 tried block ...
0
votes
1answer
36 views

the rational canonical

let T$\in$ $\mathcal{L}$($\mathbb{Q^3}$,$\mathbb{Q^3}$) be given by $$T(v)= \left[ \matrix { 1&-1&-4 \\ 1&-1&-3 \\ -1&2&-2 } \right]v $$ . Find the rational canonical form ...
2
votes
3answers
174 views

Minimizing $\|Ax\|_2$ subject to $\|x\|_2 = 1$

I have a Matlab program to estimate a vector $x$ from noisy measurements. I use the singular value decomposition (SVD) to solve the linear equation $Ax=0$ (where the number of equations is greater ...
157
votes
1answer
7k views

How does one prove the matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
1
vote
2answers
67 views

Points on which function is invertible?

$f: \mathbb R ^{2}\mapsto \mathbb R ^{2}$ $f(x,y)\mapsto((x-y)^{2}+1, x-y^{3}-2)$ For which points is this function invertible? I calculated the Jacobian matrix, but what should I do next to get ...
1
vote
3answers
68 views

Uniqueness of solution for a tridiagonal system

I have a claim I've been conjecturing. Not sure if it's true or not. Context: I'm doing some calculations with finite difference schemes. Say I have the following real $n$ x $n$ tridiagonal matrix $A$...
1
vote
2answers
49 views

For matrices $C, D$, show that $(CD)^{100} \neq C^{100} D^{100}$

The question is to prove this is false: $(CD)^{100} = C^{100}\cdot D^{100}$, where $C$ and $D$ are matrices. I looked through my textbook and could not find a proof for this.
0
votes
1answer
47 views

Matrix with all entries N

Is there a specific name for a matrix where all entries are the name number? I am writing a program where I want to be able to describe a matrix like this in the same way I would the identity matrix, ...
1
vote
2answers
70 views

How to determine which of the following matrices are similar?

If we have the following three matrices: $$ A=\begin{bmatrix} 7 &1 \\ -5 &3 \end{bmatrix},\;\; B=\begin{bmatrix} 5 &-1 \\ 1 &5 \end{bmatrix},\;\; C=\begin{bmatrix} 5 &1 \\ 1 &...
3
votes
1answer
60 views

Binary matrices with rank $n$

I'm stuck doing this problem Let $A$ be a matrix of order $n \times n$ with entries in $\{0,1\}$, which has exactly two $1$'s on each row and on each column. Which conditions are necessary and ...
3
votes
3answers
95 views

For which $a$ and $b$ is this matrix diagonalizable?

For which $a$ and $b$ is this matrix diagonalizable? $$A=\begin{pmatrix} a & 0 & b \\ 0 & b & 0 \\ b & 0 & a \end{pmatrix}$$ How to get those $a$ and $b$? I calculated ...
0
votes
2answers
47 views

Orthogonal Matrix 4 [duplicate]

Let $M_{32}$ be vector space with inner product of $AB$ given by $\text{tr}(B^TA)$. The question is to find a non-zero matrix B orthogonal to $$A=\left[\begin{matrix} 1 & 2 \\ 3 & 4 \\ 5 &...
0
votes
1answer
37 views

Orthogonal matrices 5

The question is to find in the space $\mathrm{Mat}_{3\times 2}(\mathbb{\mathbb{R}})$ a non-zero matrix that is orthogonal to $$ A= \begin{pmatrix} 1 & 2\\ 3 & 4\\ 5 & 6\\ \end{...
1
vote
1answer
88 views

$(AB-BA)^m=I_n$ has solution if and only if $n=mk$ where $m\geq 2$ is an integer number. Is it correct?

I found out that the equation $(AB-BA)^m=I_n$ does have solution when $n=km$, where $k$ is an arbitrary integer number. To prove, we just need to consider $C=$diag($r_1,..., r_m$) where $r_1,..., ...
-1
votes
0answers
57 views

$2 \times 2$ block matrix related

let $A$ be any matrix of order $n$, $J$ is matrix of order $n$ whose all entries are $1$, and $I$ is an identity matrix of order $n$, then how to find eigenvalues of following block matrix? $$M=\...
0
votes
1answer
41 views

Matrix power relation to prove coefficients exist

Let M be a 2x2 Matrix and $n$ an integer $2\geq\ n$. Prove that there exist integers $a_n$ and $b_n$ such that $$M^n=a_nM+b_nI$$ To begin, what I did was to use the Cayley-Hamilton Theorem then use a ...
0
votes
0answers
8 views

lipschitz continuity on matrix product

If $ f(t,x)=A(t)g(t,x)B(t) $ where $ A(t), g(t,x), B(t)$ represents square matrix functions. If $ A(t), B(t)$ are bounded and $ g(t,x)$ is Lipschitz continuous. Then is it correct to consider $ |f(t,x)...
0
votes
1answer
411 views

What is the matrix representation of Radon transform?

Just as the title, my question is what is the matrix representation of Radon transform (Radon projection matrix)? I want to have an exact matrix for the Radon transformation. (I want to implement ...
119
votes
5answers
5k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
2
votes
2answers
79 views

Rotation matrix check

Let matrix $A=\frac{1}{\sqrt{2}} \begin{bmatrix} 1 & -1 \\ 1 & 1 \\ \end{bmatrix} $. Check if $A$ is a rotation matrix in $\mathbb{R^2}$ by angle $\theta=\...
0
votes
1answer
31 views

For two square positive-semidefinite matrices $A$ and $B$, does the relation $|A|^{-\frac{1}{2}}|B|^{-\frac{1}{2}} = |AB|^{-\frac{1}{2}}$ hold?

For two square positive-semidefinite matrices $A$ and $B$, does the relation $|A|^{-\frac{1}{2}}|B|^{-\frac{1}{2}} = |AB|^{-\frac{1}{2}}$ hold? Here the absolute value signs are the determinant ...
0
votes
0answers
33 views

Inverse of the sum of identiy matrix and a symmetric matrix

Is there a simple way to solve $(I + A) X = B$, where $I$ is the identity matrix, and $A$ is a symmetric matrix?
1
vote
2answers
30 views

Finding the limit of a Matrices determinant

The problem is as follows: I've been trying to figure this out with no luck. I'm lost at the $A_k+1$ and $A_0$. I'm not sure what they are implying and how they would apply in finding the limit.
-2
votes
2answers
109 views

Is a correlation matrix with positive determinant PSD?

Please note: I'm not interested in the difference between positive definiteness and semi-definiteness for this question. A correlation matrix is a symmetric positive semi-definite matrix with 1s down ...
4
votes
1answer
113 views

Can I factor a rational expression of the form…

Given two equations $\displaystyle P_1 = \frac{1-X^2}{1-X^2}$ and $\displaystyle P_2 = \frac{1-aX^2}{1-bX^2}$ I am told that there is a relationship between P1 ...
2
votes
1answer
46 views

Orthogonal matrix $Q$ such that $\forall x\leq 0$, $Qx\geq 0$

What are the orthogonal matrices $Q$ such that for all vectors $x\leq 0$, $Qx\geq 0$? The inequality is to be understood component-wise. In dimension 1, the only possibility is $Q=[-1]$, which is a ...
0
votes
1answer
38 views

Does encountering a zero pivot during Gaussian elimination imply that the matrix is singular?

I was reading a problem about Gaussian elimination and pivots of a matrix, say $A$. The question is: During the Gaussian elimination process without pivoting a zero pivot has been encountered. Is ...
1
vote
2answers
37 views

Can we define component of a matrix which is orthogonal to another matrix?

Given two vectors $A$ and $B$ one can easily find component of $A$ along $B$ and component of $A$ perpendicular/orthogonal to $B$ and vice versa. This is possible as we can define dot product of two ...
3
votes
0answers
14 views

Skew-Symmetric Parts of Stochastic Matrices

It's easy to see that the set $\{W - W^T : W \in \mathbb{R}^{n \times n}\}$ is precisely the set of real skew-symmetric matrices. This continues to be the case if we restrict to (entry-wise) non-...
2
votes
2answers
82 views

Prove that the number: $z = \det(A+B) \det(\overline A-\overline B)$ is purely imaginary.

Problem: Let $A_{n\times n}$ and $B_{n\times n}$ be complex unitary matrices, where n is an odd number. Prove that the number: $$z=\det(A+B) \det(\overline A-\overline B)$$ is purely imaginary. My ...
0
votes
1answer
25 views

Basis for a matrix [closed]

Find a basis for the space M32 ? (3 by 2 matrix). I want to know basis for a matrix in general rather than being given a matrix to work with. Any help would be appreciated.
2
votes
0answers
53 views

Rotation Matrix which maps a point to an specific point

How can I compute the rotation matrix which rotates an $n$-dimensional vector $\vec{A}$ around an $n-D$ vector $\vec{O}$, and maps it to a vector $\vec{B}$ (while $\vec{A}, \vec{B}, \vec{O}$ are known)...