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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1
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0answers
17 views

a unitary relation between a matrix and its transpose

Let $F$ be a field, $E/F$ is a quadratic extension. The notions "Hermitian matrix, unitary matrix" make sense for $E/F$. Given a $n\times n$ matrix $A$, is it true that there are unitary matrix $U$ ...
0
votes
0answers
23 views

Complex square matrices, proving there exists x,y in C^n such that A=xy*

Suppose A $\in$ $M_{nxn}$ the set of complex square matrices. Show the following statements are equivalent a) A has rank 1 b) $\exists$x,y $\in$ $C^n$ such that $A=xy^*$ What are the right and ...
0
votes
2answers
25 views

Determining the dimension of the set of solutions of the system $Ax=b$ without solving for $x$ or applying row reduction

Consider the matrix $A=\begin{bmatrix}2&2&2&4\\1&2&0&-1\\1&3&-1&-4\end{bmatrix}$ part d) What is the dimension of the solution space of the homogeneous ...
2
votes
1answer
48 views

Is it possible to have an $a \times b \times c$ matrix?

The book Artificial Intelligence: A Modern Approach states that a certain variable is a $2 \times 2 \times 2$ matrix", but I thought that matrices could only be rectangular (i.e. $a \times b$). Is it ...
2
votes
3answers
46 views

Whether the set of functions $(1,e^{x},e^{-x})$ linearly independent

Are the set of functions $(1,e^{x},e^{-x})$ linearly independent? I wrote it as an augmented matrix but it brought me to nowhere. Can somebody help me?
-2
votes
0answers
29 views

Standard matrix for reflection about the plane $x=z$ followed by shifting [on hold]

Determine the 3x3 standard matrix of a linear transformation that permutes and translates the components of a 3-dimensional vector as $v = (v_{x}, v_{y}, v_{z})$ $T(v) = (v_{z}+1, ...
1
vote
3answers
91 views

Prove/disprove that the matrix $AB+BA=0$

Let $A,B$ be matrices of an order $5\times 5$ such that: $$AB+BA=0$$ Prove/disprove: at least one of the two matrices is not invertible. I wanna say I tried various ways before I posted here but I ...
0
votes
2answers
25 views

Express the polynomial $ax^2+2hxy+2gx+2fy+by^2+c$ in matrix notation

I'm given $$\begin{bmatrix}x & y & 1\end{bmatrix}*M*\begin{bmatrix}x \\ y \\ 1\end{bmatrix}$$ where $M$ is the polynomial $ax^2+by^2+2hxy+2gx+2fy+c$ in matrix notation. Im totally stumped ...
0
votes
1answer
7 views

Number of bit operations in nxn zero-one matrix boolean product

I was reading transitivity closure from the book Discrete Mathematics and Its Application by Kenneth Rosen It says that in the boolean product of nxn zero-one matrix, there are $n^2(2n-1)$ bit ...
3
votes
1answer
55 views

Determining $\det(\mathbf{A})$ using the characteristic polynomial

Let the 3x3 matrix be $ \mathbf{A} = \begin {bmatrix} 3&1&0\\1&3&0\\0&0&1 \end {bmatrix}$. a) Determine its eigenvalues and eigenvectors. b) Do the eigenvectors ...
0
votes
0answers
22 views

A simple matrix proof [duplicate]

Let $A$ be a matrix of an order: $(n\times n)$, Prove/disprove: $AC=CA$ for every invertible matrix $C$ of an order $n\times n$ if and only if: $$ a_{ij} =\begin{cases} c \in R, & i=j \\ 0 ...
5
votes
1answer
31 views

what can be derived from similar matrix

If $A=\begin{pmatrix} 0&\star&\star \\ \star&x&\star \\ \star & \star & 5 \end{pmatrix}$ is similar to $B=\begin{pmatrix} 1&0&0 \\ 0&y&0 \\ 0 & 0 & 10 ...
0
votes
0answers
22 views

update cholesky factorization

I need to compute cholesky(H'*H) where H is a big sparse rectangular matrix. After that H is modified by adding several lines. That is Hn = [H ; line_1 ; ... ; line_n] in Matlab. How can I recompute ...
3
votes
0answers
20 views

An inequality concerning restricted isometry property

Let $A\in \mathbb{R}^{m\times n}$ be a matrix and let us denote by $A_S$ the submatrix of $A$ with the columns restricted to a set $S\subset [n]:=\{1,2,\cdots, \ n\}$. Then one says that the matrix ...
2
votes
1answer
40 views

$n\times n$ matrix with all eigenvalues equal to $1$ or $0$. Does a conjugated matrix with only $1$'s and $0$'s exist?

Let $A$ be an $n\times n$ matrix with all eigenvalues equal to $1$ or $0$. Is there a conjugated matrix $B = XAX^{-1}$ for some $X$ such that all the elements equal either $1$ or $0$? My thoughts so ...
1
vote
3answers
30 views

How close apart are two message - “Document Distance” algorithm

I was looking at this algorithm that computes how close apart are two texts from one another and the formula seems a bit weird to me. The basic steps are: For each word encountered in a text you ...
0
votes
1answer
22 views

Shifting of the spectrum of a linear operator - in both the symmetric and non-symmetric cases,

a) I finished a problem that sort of highlighted the fact that if a real symmetric matrix $A_2$ = A + I, where A is also real and symmetric, then $A_2$ has the same eigenvectors as A, but its spectrum ...
0
votes
1answer
20 views

Align the cube's nearest face to the camera

I have a cube and 4x4 transformation matrix Cube is rotated randomly I need to find the nearest face of cube regarding to camera and rotate the cube by aligning that face to the camera. How can I do ...
0
votes
2answers
27 views

Square block matrix, with Hermitian, non-negative definite blocks, prove that the matrix is also non-negative definite,

Consider the square block matrix $$S= \begin{bmatrix} R & RQ^* \\ QR & QRQ^* \\ \end{bmatrix} $$ where $R$ is a Hermitian, non-negative definite square matrix ...
3
votes
1answer
82 views

I can't understand a step in the proof of the associativity of matrix multiplication

Matrix multiplication is proven by the following reasoning: Let there be matrices $A^{m \times n}$, $B^{n \times k}$ and $C^{k \times l}$. Then $$ ...
3
votes
1answer
24 views

Calculating a bound on the norm of a matrix exponential

The problem is this: Let A be a square $n \times n$ matrix, and define $$e^A=\sum_{k=0}^\infty \frac{1}{k!}A^k$$ Find a bound for $\lvert e^A \rvert$ in terms of $\lvert A \rvert$ and $n$. I was ...
0
votes
1answer
27 views

Over-specified linear system

Consider the matrix $A $ with RREF consisting of three of the 4, 4- dimensional standard vectors: $[\mathbb {e_1}, \mathbb {e_2}, \mathbb {e_3} ] $ Since the rank is 3 the matrix has one solution ...
1
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0answers
27 views

matrix with positive diagonal elements

I was wondering if a symmetric matrix with positive elements only in the diagonal (negative elsewhere) is any special beside the symmetry. Thanks in advance
0
votes
1answer
13 views

Generation of rank-$2$ matrices from a dictionary of rank-$1$ matrices.

I have a question about the construction of rank-$2$ matrices from a dictionary of rank-$1$ matrices. Consider the set $\mathcal{D} = \{\mathbf{A} \in \mathbb{C}^{2 \times 2} \mid rank(\mathbf{A}) = ...
2
votes
1answer
30 views

how to find the dimension of the image of $f$ in this case?

Let $A \in M_{m \times n}(\Bbb R)$ be fixed, and let $B \in M_{m \times l} (\Bbb R)$. Consider the map $f: M_{n \times l}(\Bbb R) \to M_{m \times l}(\Bbb R)$ defined by $f(X) = AX + B$ for all ...
1
vote
1answer
23 views

Determining the standard matrix from the images of the standard basis vectors

Let a linear transformation $T:$ $\mathbb{R}^3$ → $\mathbb{R}^3$ rotate a vector around the z-axis by $45^{o}$ followed by an orthogonal projection onto the x-axis. Determine the standard matrix ...
0
votes
2answers
31 views

Relationship between eigenvalues of A symmetric matrices

Let $$A=\begin{pmatrix}a & b\\b & c\end{pmatrix} \in M_2\mathbb{(R)}$$ i) Find the eigenvalues of $A$ ii) If $\begin{pmatrix}1\\2\end{pmatrix}$ is an eigenvector of $A$, prove that ...
0
votes
0answers
10 views

Efficient inverses of many related matrices [duplicate]

Say I have a $N$-by-$N$ positive definite real matrix $\Sigma$ and I wish to compute the inverses (or equivalently Cholesky decompositions) of $(\Sigma + a_k I)^{-1}$ for a set of $K$ positive $a_k$. ...
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votes
0answers
14 views

Finding bases for kernel N(T) and image R(T)?

Goodday, I need some help with the following problem: Find bases for both N(T) and R(T) in the following transformation: $T: M_{2 \times 3}(F) \rightarrow M_{2 \times 2}(F)$ defined by: ...
1
vote
2answers
22 views

Inverse of nonnegative Toeplitz matrice

Consider a right-hand circulant matrice of size $n$ (called also Toeplitz matrice) \begin{equation} T= \left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_n \\ a_n & a_1 & ...
1
vote
2answers
64 views

Simultaneously diagonalization of two matrices.

Let $A$ be a real symmetric matrix and $B$ a real positive-definite matrix. Is it possible to simultaneously diagonalize of $A$ and $B$? Thank you very much.
2
votes
1answer
30 views

stuff involving adjoint, self adjoint [on hold]

Let $T: V \to V$ be a linear transformation relative to a finite dimensional Euclidean space $V$ (real or complex). Prove that there exists linear transformation $T^*: V \to V$ (called the adjoint ...
6
votes
1answer
68 views

Eigenvalues of symmetric matrix with skew-symmetric matrix perturbation

If $A$ is diagonalizable, using the Bauer-Fike theorem, for any eigenvalue $λ$ of $A$, there exists an eigenvalue $μ$ of $A+E$ such that $|\lambda-\mu|\leq\|E\|_2$ (the vector induced norm). Here I ...
0
votes
4answers
74 views

$A.A^t$ is diagonal

Be $A$ a semidefinite nonnegative matrix. What kind of conclusions can we say about $A$ if $A.A^t$ is diagonal? Same question when $A$ is binary matrix. Thanks
6
votes
1answer
63 views

If $A$ and $B$ are $n×n$ matrices such that $AB=B$ and $BA=A$ then find the value of $A^{4} + B^{4} - A^{2} -B^ {2} + I$

The given question is If $A$ and $B$ are $n×n$ matrices such that $AB=B$ and $BA=A$, then find the value of $A^{4} + B^{4} - A^{2} -B^ {2} + I$. Any hints?
6
votes
3answers
110 views

Prove that $\det(A^2 + A + xI) = x$

Let $x$ be a positive real number and $A$ a $2\times2$ matrix with real values satisfying the following property $\det(A^2 + xI) = 0$. Prove that $\det(A^2 + A + xI) = x$ I have tried something with ...
0
votes
2answers
35 views

Finding all matrices for which the homogeneous system has a given solution space

Find all $3\times 3$ matrices for which the homogeneous system has a solution space as the line $x = 2t$, $y = t$, $z = 0$. (Hint: Write the row reduced augmented matrix from given information.) ...
0
votes
3answers
27 views

Rank of matrices and their product

Let $\operatorname{rank}(A_{3 \times 3})=\operatorname{rank}(B_{3 \times 3})=2$. I need to figure out whether $AB=0$ is possible. On the one hand, I know that $\operatorname{rank}(AB) \leq ...
3
votes
2answers
58 views

Given $A$ is $6×6 $ real symetric matrix of rank $5$ , then to determine rank of $A^{2}+ A+I $

Given $A$ is $6×6 $matrix of rank $5$ , then to determine rank of $A^{2}+ A+I $. I knowthat rank of matrix doesnot change when we square it , but how to proceed in this question.Any hints ? Thanks
0
votes
0answers
24 views

Solving this matrix trace equation

We have that $D \in \mathbb{R}^{n \times n}$ is a diagonal matrix, $P \in \mathbb{R}^{n \times n}$ an orthogonal matrix and $V \in \mathbb{R}^{n \times n}$. Also $\psi \in [0,1] \subset \mathbb{R}$ ...
5
votes
4answers
185 views

Proof If $AB-I$ Invertible then $BA-I$ invertible.

I have these problems : Proof If $AB-I$ invertible then $BA-I$ invertible. Proof If $I-AB$ invertible then $I-BA$ invertible. I think I solve it correctly, But I'm not so sure, I'll be glad to ...
0
votes
1answer
38 views

Derivative of a matrix with respect to a scalar

I would really appreciate some help in finding the partial derivative of the following with respect to $\psi$: \begin{equation} \mbox{trace}((XX^\top)^{-\psi}V) \end{equation} Here, $\psi \in [0,1] ...
0
votes
0answers
10 views

How to compute the unique positive eigenvector 'v' in Analytic Hierarchy Process

I'm trying to calculate the values in the right most column v but I have absolutely no idea how to do it. I've done some prelim work and managed to get pretty much everything else in the table in ...
1
vote
3answers
26 views

Matrix multiplication ambiguity

From this source here, it says that matrix multiplication is given by this: $AB = \begin{bmatrix} a_{1,1}b_{1,1}+a_{1,2}b_{2,1}+...+a_{1,n}b_{p, 1} & ...\\ \vdots & ...
0
votes
0answers
15 views

Multiplications of non-square matrices and the dependencies of row vectors.

I'd like to find $D$ and $L$ for a given $H$. $H$ is a 7-by-6 matrix. Its rank is 6. All sub-matrices of $H$ are full rank. In other words, if we choose any $n$-by-$n$ sub-matrix within $H$, where $n ...
0
votes
1answer
28 views

How do row operations affect the column space?

I've been curious about this: Row operations do not affect the row space, but they affect the column space. Is there any way to 'systematically' perform row operations to make the column space the ...
4
votes
1answer
63 views

Matrix equation solution

Does anybody know how to solve this matrix equation: $$P = P P^T R + X,$$ where $P, R,$ and $X$ are vectors with $n$ elements, and $P$ is the unknown vector?
0
votes
1answer
21 views

Relationship between type of matrix and eigenvalues

Prove that if the eigenvalues of a diagonalizable matrix $A\in M_n(\mathbb{R})$ are all $1$ or $-1$, then $A^{-1}=A$ What I tried to reverse the way to get the rough idea. $$A^{-1}=A\implies ...
-1
votes
1answer
53 views

True / False about a matrix

Let $A= \begin {pmatrix} x & y \\ -y & x \end {pmatrix}$ where $x,y \in \mathbb{R}$ such that $x^2+y^2=1$. 1) For any $n \ge 1$, $$A^n= \begin {pmatrix} \cos\theta & \sin \theta \\ -\sin ...
1
vote
2answers
39 views

Eigenvalues of $6 \times 6$ matrix?

Which of {$\pm1,\pm i$} are the eigenvalues of matrix A, $$A=\begin{pmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & ...