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
votes
1answer
12 views

Find orthonormal bases for the orthogonal subset

Let S be the subset $$\{\left(1,0,2\right),\left(3,2,1\right),\left(1,-2,7\right)\}\subset\mathbb{R}^{3}$$ Find orthonormal bases for $S^{\perp}$ and $S^{\perp\perp}$ I have begun by putting these ...
1
vote
1answer
19 views

Symmetric, Antisymmetric, and Alternating Bilinearforms form a vector subspace

I have to show that the space of symmetric, the antisymmetric and the alternating bilinear forms each form a vector subspace of the space of all bilinear forms $\operatorname{Bil}(V,K)$ with $V$ being ...
7
votes
2answers
383 views

Are linear combinations of powers of a matrix unique?

One can see from the Cayley-Hamilton Theorem that for a $n\times n$ matrix, we can write any power of the matrix as a linear combination of lesser powers and the identity matrix, say if $A\neq cI_n$, ...
0
votes
0answers
21 views

$X \subseteq M(n,\mathbb C)$ and $S:=\{x \in \mathbb C : x$ is an eigen value of some matrix in $X\}$ ; does $X$ closed implies/if $S$ is closed?

Let $X \subseteq M(n,\mathbb C)$ and $S:=\{x \in \mathbb C : x$ is an eigen value of some matrix in $X\}$ ; does the closed-ness of any one of $S$ or $X$ implies that the other set is also closed ?
-1
votes
1answer
37 views

Kernel of linear transformation [closed]

I was trying to help my son on his math assignment , I have faced a tricky questions , I will highly appreciate if someone can guide me in this task , because the questions are really confusing me ...
0
votes
1answer
21 views

Every matrix is congruent to a diagonal matrix

I think I've read somewhere that any square matrix $M$ can be decomposed as $M=P^TDP$, where $D$ is a diagonal matrix. Such statement looks very much like the spectral theorem, although in this case ...
3
votes
1answer
46 views

Help with understanding the proof for: $AB$ and $BA$ have the same characteristic polynomial (for square complex matrices)

I saw many proofs but they all use advanced techniques and are impossible to understand. I'm looking for a proof that $AB$ and $BA$ have the same characteristic polynomial for any square matrix over ...
2
votes
2answers
56 views

Diagonalisation proof

Suppose the nth pass through a manufacturing process is modelled by the linear equations $x_n=A^nx_0$, where $x_0$ is the initial state of the system and $$A=\frac{1}{5} \begin{bmatrix} 3 ...
1
vote
1answer
41 views

$GL_2(\Bbb C)$ acts on a certain set

Let $G:=GL_2(\Bbb C)$, $B$ and $T$ be the subgroup consisting of all upper triangular and diagonal matrices in $G$, respectively. Set $w:= \left( \begin{array}{cc} 0 & 1\\ 1 & 0 ...
3
votes
3answers
66 views

$\operatorname{trace}(AB) = 0$ and $\operatorname{rank} (A)=1$. Prove: $ABA=0$

I know that $AB-BA=A \iff$ $A$ is singular. $A$ and $B$ can be complex. Any hints?
1
vote
1answer
15 views

For integers $n>1$ , $k$ , does there exist matrix $A$ with integer entries and first row $(1,2,…,n)$ such that $\det A=k$?

Let $n >1$ be an integer , then is it true that for any integer $k$ , there exist a matrix $A \in M(n,\mathbb Z)$ with first row of $A$ as $(1,2,...,n)$ such that $\det A=k$ ?
0
votes
0answers
20 views

Matrix block with Schur Complement

Consider $X\in S^n$, $S^n$ is a space of $n\times n$ symmetric matrices, partitioned as $$X = \begin{bmatrix}A & B \\B^T & C\end{bmatrix}$$ where $A\in S^k$. If det $ A\neq0$, the matrix ...
0
votes
0answers
5 views

Can I always find 2 independent axises from data distributed in x, y panel?

Let's say, I have data (x, y) distributed at x,y panel: Can I always find a pair of (u, v) axis, so that data along these axis are independent from each other? u, v are just x, y rotate at ...
0
votes
0answers
104 views

Prove that $ \det{\begin{bmatrix}A & B \\-B & A\end{bmatrix}}\geq 0$ [duplicate]

Let $A,B \in M_n(\mathbb{R})$. Prove that $\det{\begin{bmatrix}A & B \\-B & A\end{bmatrix}}\geq 0$. I know that there is a theorem which says that if $E,F,G,H \in M_n(\mathbb{F})$ and ...
7
votes
2answers
262 views

Non-negative determinant of a block matrix

Here's the problem I've been stuck on for some time now. Let $A,B \in M_n(\mathbb{R})$. Let $C= \begin{bmatrix} A & B \\ -B & A \\ \end{bmatrix} $ be a real ...
-3
votes
0answers
13 views

Find $A$ such that $Av$ has only non-zero entries if $v$ has only non-zero entries [closed]

I am trying to find all real matrices $A$ of dimension $n$ such that if $v$ is an arbitrary column with all entries non-zero then all entries of column the $Av$ are non-zero too.
1
vote
1answer
19 views

Find a $3 \times 3$ matrix $A$ such that $g(x) = AxT$

I am new to linear transormations and I can do the $\Bbb R^m \to \Bbb R^n$ tranformations, however I came up to this question and im confused. I'd be grateful if any of you could guide me how to start ...
-2
votes
0answers
29 views

Linear algebra , magical square? [closed]

Let $T:\mathbb{R} ^{3}\rightarrow \mathbb{R} ^{3}$be a linear transformation . Prove the equivalance following statements : i) $\mathbb{R} ^{3}=ker\left ( T\right) \oplus im\left ( T\right)$ ii) ...
0
votes
0answers
19 views

Create a matrix, smallest size possible [closed]

(A) Create a matrix, smallest size possible, where α ±i β is a pair of complex-conjugate eigenvalues of multiplicity 3 with defect = 1 and λ is a real eigenvalue of multiplicity 3 with a 1- ...
1
vote
1answer
18 views

Covariance matrix - wrong results either in Matlab or in formula?

The definition of a covariance for discrete variables (taken from https://en.wikipedia.org/wiki/Covariance) is: $$\operatorname{cov} (X,Y)=\frac{1}{n}\sum_\limits{i=1}^n (x_i-E(X))(y_i-E(Y))$$ ...
4
votes
1answer
58 views

Inverting an $n \times n$ matrix using determinant

We're asked to invert the following matrix with the help of guided questions. $$\begin{pmatrix} 1 + a_1 & 1 & \cdots & 1 \\ 1 & 1+a_2 & \ddots & \vdots \\ \vdots & \ddots ...
1
vote
2answers
25 views

Dimension of solution space of homogeneous system of linear equations

I have the homogeneous system of linear equations $$ 3x_1 + 3x_2 + 15x_3 + 11x_4 = 0, $$ $$ x_1 − 3x_2 + x_3 + x_4 = 0, $$ $$ 2x_1 + 3x_2 + 11x_3 + 8x_4 = 0. $$ I have converted to a augmented ...
0
votes
1answer
31 views

Inverse of a 3x3 matrix with 4 0's in the non diagonals

Hey how do I find an inverse for a matrix in the form \begin{bmatrix}a&0&b\\0&c&0\\d&0&e\end{bmatrix} without having to use traditional methods, in the exam we are expected to ...
2
votes
1answer
37 views

Does this matrix series have an answer?

I'm trying to solve this series: $$\displaystyle\sum_{i=0}^{k}A^i B C^{k-i}$$ Where A, B, and C are $N\times N$ symmetric matrices. And $A$ and $C$ have spectral radii smaller than or equal to 1, ...
1
vote
2answers
48 views

Find a basis for the image and kernel of a linear transformation

I'm having a bit of difficulty obtaining an answer to this problem. Specifically, finding a basis for the kernel of a transformation, $\ker(T)$ . Let \begin{align*} p_1(t)&=-1+t \\ ...
0
votes
1answer
19 views

Simultaneous Diagonalization of A and B via $\Sigma = A^{-1}B$

I am reading the paper "A Generalized Subspace Approach for Enhancing Speech Corrupted by Colored Noise" by Yi Hu and Philipos C. Loizou. In the paper, they claim that given two matrices $R_{n}$ and ...
0
votes
0answers
15 views

Rank of adjoint of a matrix [duplicate]

I need to prove these 3 statements, and I don't know how to start... A is an nxn matrix: 1) if rank(A) = n then rank(adj(A)) = n 2) if rank(A) = n-1 then rank(adj(A)) = 1 2) if rank(A) < n-1 ...
1
vote
1answer
89 views

Determinant of augmented matrices.

Let $A$ and $B$ be $n \times n$ real matrices. How can I show that $\det \begin{bmatrix} A & B \\[0.3em] -B & A \\[0.3em] \end{bmatrix} \geq 0 $?
1
vote
1answer
24 views

$X \subseteq M(n,\mathbb C) ; |X|>1 ; $ connected /path connected , what about $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix in $X\}$?

Let $X \subseteq M(n,\mathbb C)$ be a set with more than one element ( I am also considering $M(n,\mathbb R) \subseteq M(n,\mathbb C)$ and $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix ...
2
votes
3answers
40 views

Let $A$ be an $n\times n$ matrices over $\mathbb{C}$ then which of the following are true?

Let $A$ be an $n\times n$ matrix over $\mathbb{C}$ such that every non zero vector of $\mathbb{C^n}$ is an eigenvector of $A$. Then, All eigen values of $A$ are equal. All eigen values of ...
0
votes
0answers
10 views

How to derive the procedure for scatter matrix

I am studying about the multiple discriminant analysis and I am suffer from the matrix calculataion. I think it is so easy, but it is not easy for me I am welcome all of you hints and comments and ...
3
votes
3answers
69 views

If $B^2$= this matrix, find $B$

Find the real matrix $B$ such that $$ B^2 = \begin{pmatrix} 2 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2 \end{pmatrix} $$ I think I am meant to use egenvalues to solve this, but would ...
1
vote
1answer
43 views

Derivatives using matrices good

$$\left|\begin{matrix} (1+x)^{a_1b_1} & (1+x)^{a_1b_2} & (1+x)^{a_1b_3} \\ (1+x)^{a_2b_1} & (1+x)^{a_2b_2} & (1+x)^{a_2b_3} \\ (1+x)^{a_3b_1} & (1+x)^{a_3b_2} & (1+x)^{a_3b_3} ...
0
votes
2answers
32 views

Eigenvalues of $A=\mathbf{u}_1\mathbf{u}_1^T + \mathbf{u}_2\mathbf{u}_2^T$

Let $\mathbf{u}_1, \mathbf{u}_2\in \mathbb{R}^{N\times 1}$ be real-valued vectors. Assume that $\mathbf{u}_1$ is orthogonal to $\mathbf{u}_2$ (i.e., $\mathbf{u}_1^T\mathbf{u}_2=0$). Let us consider ...
1
vote
1answer
21 views

Find all possible Jordan Canonical form for nilpotent matrices with the characteristic polynomial $\lambda ^6$

As far as I understand the way to solve it is to go from $(m_A=\lambda^k)_{k=1}^6$, the largest block will be of size $k$ and find the sizes that the rest of the blocks can be by finding ways to add ...
1
vote
3answers
42 views

Find a matrix $A \in \mathbb{R}^{2 \times 2}$ such that $ \|Ax\|_{2}=\|x\|_{2}$ for every $ x\in \mathbb{R}^2 $

How to find a matrix $A \in \mathbb{R}^{2\times 2}$, $A\neq I_{2}$ such that for every $ x\in \mathbb{R}^2$ we have $\|Ax\|_{2}=\|x\|_{2}$. Is that even possible?
2
votes
1answer
41 views

Connectedness of the sets $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) =m\}$ and $\{A \in M(n,\mathbb R) : \mathrm{rank}(A)\ge r\}$

Let $r>0$ , I know that $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) < r\}$ is path connected in $M(n,\mathbb R)$ . My question is ; for positive integer $m < n$ , is the set $\{A \in ...
2
votes
1answer
27 views

Under what conditions does $AEA^{-1}=E $

In matrix multiplication when does $AEA^{-1}=E $? if any are the identity then it's trivial. Extra question, are there non diagnol matrices solutions or a proof one can't exist?
5
votes
3answers
45 views

Find a matrix $E$ such that $EA= B$

I am asked to find a matrix $E$ such that $EA= B$. I am given matrix $A$ which is $4\times 4$ and matrix $B$ $4\times4$. Would I find $E$ the following way or is incorrect? $$EA=B$$ $A^{-1} [EA = ...
0
votes
1answer
43 views

Kernel of a polynomial

Let's say we have a 2nd degree polynomial $a+bx+cx^2$ and it is given that $T:P2\rightarrow R$ given by $T(p)=\int_{0}^{1}p(x)dx$ We are asked to find the kernel of $T$. Now, I know that depending ...
1
vote
2answers
53 views

Properties of a $3 × 3$ matrix $A$ that contains two equal rows.

A $3 × 3$ matrix $A$ contains two equal rows. State whether each of the following is true or false. (a) $A$ has an inverse. (b) The rows of $A$ are linearly independent vectors. (c) The determinant ...
2
votes
2answers
51 views

For which $\lambda$ do we have solutions

I'm trying to find for what values of $\lambda$ the following matrix has either no solutions, infinitely many or unique solutions. $$A=\begin{pmatrix} 1 & 1 & \lambda & 1 \\ 4 & ...
1
vote
1answer
15 views

Which one is the correct definition of natural norm?

In the definition 2 of Normal Subgroup Reconstruction and Quantum Computation Using Group Representations, the authors have defined the natural norm of a matrix as follows. The natural norm of the ...
1
vote
1answer
18 views

What is the “default” direction of feature / parameter vector?

This seems like a pretty obvious thing, so it's never really explained, but I can't understand it. Many book chapters use the expression: $\boldsymbol w^{T}\boldsymbol x$ as a form of denoting ...
1
vote
1answer
53 views

Can any linear transformation be represented by a matrix?

Use $\cal L$ to denote a linear transformation on some vector space. We know any matrix $\bf{A}$ can be viewed as a linear transformation by defining $\cal L:= \cal L(\bf{v})= Av$ where $\bf{v}$ is a ...
2
votes
0answers
21 views

Show that matrix is totally unimodular

I want to show that this matrix is totally unimodular: \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & ...
1
vote
1answer
32 views

What is a particular use of Gram-Schmidt orthogonalization?

We have a linear space V of m x n matrices. I know that we can use Gram-Schmidt to construct an orthonormal basis but the natural basis for this space (where every ij-th element is 1 and the rest 0) ...
-1
votes
0answers
10 views

Property or feature of a symmetric or skew symmetric matrix for comparison

I have a symmetric matrix of size $n\times n$, where $n$ may range from $10$ to $50$. Let us call this matrix $M_1$. If from $M_1$, I delete an $m$th row and an $m$th column, all other elements of ...
0
votes
0answers
14 views

Property / Feature of a symmetric or skew symmetric matrix for comparison

I have a symmetric matrix of size $n \times n$, where $n$ may range from $10$ to $50$. Let us call this matrix as $M_1$. If from $M_1$, I delete an $m$th row and an $m$th column, all other elements ...
0
votes
0answers
12 views

Cholesky of a special block matrix using cholesky of sub matrix

Let $A$ be a real $N\times N$ positive semi-definite matrix. Let $r$ be a real $N\times 1$ vector. Then consider the matrix \begin{align} B = \begin{bmatrix}A & r \\ r^T & 0\end{bmatrix} ...