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
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1answer
229 views

Why is this matrix multiplication identity true?

Let $X$ be an $n \times p$ matrix, and let $\mathbf 1$ be a vector of $1$'s of length $n$. Why does the following hold (assuming $X'X$ is not singular): $$\left( X'X \right)^{-1}X' ...
2
votes
1answer
524 views

determinant of a Hankel matrix

This is a question on Hankel determinant. Is there a closed form for determinant of the Hankel matrix of the sequence $\{1, a, a^2,\cdots, a^{2n+2}\}$?
2
votes
1answer
276 views

Similar matrices and rref

I have a hunch that all conjugate/similar matrices have the same reduced row echelon form. Am I right?
2
votes
2answers
156 views

Showing $AS$ is orthogonal if $A$ and $S$ are orthogonal

Suppose matrix $A$ is an $n \times n$ orthogonal matrix and $S=\{\vec{u}_1, \vec{u}_2, \ldots, \vec{u}_n\}$ where $\vec{u}_1, \vec{u}_2, \ldots, \vec{u}_n$ are orthogonal to each other. Now, since ...
2
votes
1answer
668 views

A question on linear transformation

Let $T$ be a linear transformation on $\mathbb{R}^{4}$ whose standard matrix is $$\left(\begin{array}{rrrr} 1 & -1 & -1 & -1\\ 1 & 1 & 1 & -1\\ 1 & 1 & -1 & 1\\ ...
2
votes
2answers
2k views

Subordinate matrix norm

I have the following matrix norm: $$\Vert A \Vert = \max_{1\leq i, j\leq n} \vert a_{ij} \vert \>.$$ I have to decide if this is a subordinate matrix norm or not. I have tried to use the ...
2
votes
1answer
523 views

An example of a linear transformation that is not 1-1 or onto but has bijective dual

This is a question from some ring theory problems in Linear Algebra. I am not sure if this question is simple enough to come up with a linear map on a finite dimensional vector space but here it ...
2
votes
1answer
152 views

Get matrix coords from a single index and matrix dimensions

Having a set of ordered numbers $$ \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 \} $$ If you put them in a matrix so the index is at the same time the content Matrix dimension $=$ row x col $= 3\times ...
2
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1answer
135 views

Real square matrices space as a manifold

Let $\mathrm{M}(n,\mathbb{R})$ be the space of $n\times n$ matrices over $\mathbb{R}$. Consider the function $m \in \mathrm{M}(n,\mathbb{R}) \mapsto (a_{11},\dots,a_{1n},a_{21}\dots,a_{nn}) \in ...
2
votes
1answer
1k views

Determining the Jordan form of a matrix given the invariant factors

I am trying to recover the Jordan normal form of a matrix given a list of invariant factors and was wondering if I am proceeding correctly in constructing the Jordan blocks. Let $F = \mathbb{C}$ and ...
2
votes
2answers
552 views

Inverse of an upper-left triangular (partitioned) matrix

I'd appreciate help finding the inverse of the upper-left triangular (partitioned) matrix $$ \left[ \begin{array}{ll} \mathbf{K} & \mathbf{P} \\ ...
2
votes
2answers
226 views

Always solving systems of linear equations wrong - what do I do wrong?

Dear ladies and gentlemen, over time I noticed I (and other) again and again have problems solving "systems of linear equations". It seems depending of the steps one chooses, we get different ...
2
votes
2answers
103 views

Sparseness for a matrix

I would like to define a function $f$ whose range is $[0,1]$ such that it takes a matrix $C \in R_+$ of dimension $m \times n$. The entries in the matrices are also in the range $[0,1]$. In addition, ...
2
votes
1answer
558 views

Least squares approximation (matrices)

Hey! I’m quite stuck on the question below. It keeps coming up in my exam and I don’t know how to do it! Please help me! Thanks! Show that the matrix $P = {A (A^tA)^{-1} A^t}$ represents an ...
2
votes
2answers
558 views

Transformation and Matrices - Points and Vectors

Right question, I am stuck. We have been working on matrices and I think I understand them, however I have no idea how to apply these to this transformation question. Consider the points O = (0; 0; ...
2
votes
3answers
510 views

What are the dimensions of the product of two matrices?

A simple question is a (5x2)*(2x5) = a (5x5) matrix?
2
votes
1answer
916 views

Linear Regression with 3x3 Matrices

Here's my Homework Problem: We can generalize the least squares method to other polynomial curves. To find the quadratic equation $y=a x^2+b x+c$ that best fits the points $(-1, −3)$, $(0, 0)$, ...
2
votes
1answer
343 views

Bounds on inverse elements of Hermitian matrices

Let $A$ be an $N$ by $N$ Hermitian matrix with elements $a_{ij}$. What will be the bound on the elements $b_{ij}$ where $B=A^{-1}$? If $A$ is a diagonal matrix, solution is trivial. Also for ...
2
votes
1answer
1k views

transpose of positive matrix is positive

how to prove it? I am talking about matrixes which: ( Ax , x ) > 0 for any x not equals 0. How to prove, that AT is also positive? xT * A * x = ( xT * A * x )T and what?
2
votes
2answers
21 views

Eigenvalue of a matrix and a polynomial of that matrix

Let $A$ be a $n \times n$ matrix over $F$, and let $c_1, ... c_n$ be its eigenvalues. Show that for every polynomial $g(x) \in F[x]$, the eigenvalues of $g(A)$ are $g(c_1), ... , g(c_n)$. I think by ...
2
votes
2answers
62 views

Eigenvalues and eigenvectors of the Householder matrix $H = I - \frac{2}{u^Tu} uu^T$

So during my first revision for the semester exams, I went through exercises in books/internet and I found 2-3 that caught my eye. One of them was the following: Let $u \in \mathbb R^n$ be a ...
2
votes
1answer
65 views

Prove that $R(+,.)$ is a division ring but I disproved it

QUESTION: Let $R=\left[\begin{matrix}\alpha & \beta \\ \bar\beta & \bar\alpha\end{matrix}\right]\in \mathbf{M_2(\mathbb{C})} $ where $\bar\alpha,\bar\beta$ denote the conjugates ...
2
votes
2answers
32 views

How do I find value of a and b in this matrix question?

This is a question from a homework sheet my teacher gave. I already did alternate a. Alternate b is quite confusing! It asks to find the value for a and b. I don't really know what to do but here's ...
2
votes
1answer
21 views

Orthogonal matrix representation

If $\mathbf{M}$ is anti-symmetric, then $\mathbf{U}=(\mathbf{I}-\mathbf{M})(\mathbf{I}+\mathbf{M})^{-1}$ is orthogonal with $\det\mathbf{U}=1$. This is just manipulation and noticing that ...
2
votes
2answers
58 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 ...
2
votes
1answer
36 views

What Similarity of Matrices really mean?

If matrices are similar then what in layman language it would imply? Are there some properties of matrices which we would expect to be similar?
2
votes
2answers
33 views

For which complex parameters the following matrix is diagonalizable

For all possible complex values of the parameter $\lambda$, determine if the matrix $A$ is diagonalizable and if so find an invertible matrix $C$ and a diagonal matrix $D$ so that $C^{-1}$$DC=A$ $A$ ...
2
votes
3answers
56 views

Finding conditions on the eigenvalues of a matrix

Consider the $2\times2$ matrix $$A = \begin{pmatrix}a&b\\c&d\end{pmatrix}$$ where $a,b,c,d\ge 0$. Show that $\lambda_1\ge\max(a,d)>0$ and $\lambda_2\le\min(a,d)$. So the eigenvalues ...
2
votes
2answers
55 views

linear algebra (matrices) - challenging problem (determination of method/algorithm)

I wonder about the following method/algortithm about square matrices $A_{n \times n}$ $\in$ $M_{n\times n}(\mathbb{K})$, where $\mathbb{K} $ $\in$ {$ \mathbb{R}, \mathbb{C}$ }. Given certain value of ...
2
votes
2answers
42 views

Do four points lie on the circumference of a single circle? Can I solve this with matrices?

I think I managed to figure out a way to determine whether three points lie in a single line via matrix determinants (but correct me if there's a problem): Where $y - mx - b = 0$, I plug each of the ...
2
votes
1answer
17 views

Determining complex eigenvalues problem

Im solving the eigenvalue for my Matrix A with eigenvalue: $2i$ -2i 0 2 0 2-2i 0 -2 0 -2i This reduces to: ...
2
votes
2answers
42 views

Known classes of Hadamard matrices

In the book Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices by Wallis et al., Appendix A of the chapter on Hadamard matrices gives a list of known classes of Hadamard matrices. However, ...
2
votes
1answer
41 views

Determine if a matrix can be transformed to a nonnegative matrix

Non-negative matrices come up in relation to the Perron-Frobenius theorem. The definition of a nonnegative matrix is that all of the matrix elements are greater than or equal to zero. From this, the ...
2
votes
2answers
39 views

What happens to dimension of kernel when squaring non-invertible matrix?

I need to show that given $A \in \mathbb{M}_{(3 \times 3)}$ $A^2 \neq [0]$ and $A^3 = [0]$, there exists $\vec{v}$ s.t. $\{\vec{v}, A\vec{v}, A^2\vec{v}\}$ is a basis in $\mathbb{R}^3$. I was hoping ...
2
votes
2answers
39 views

Determinat of block matrix related

How to find determinant of following block matrix? $\begin{bmatrix} A & A \\ A & kI \end{bmatrix}$ Where $A$ is any square matrix,$I$ is an identity matrix and $k$ is any constant.
2
votes
2answers
38 views

How many arithematic operations(flops) are to $n×(n+1)$ matrix of system?

Source: Linear Algebra and Its Applications David C. Lay A system of n equations in n unknows correspond to $n×(n+1)$ augmented matrix. One book says the reduction(elimination) to echelon form ...
2
votes
1answer
74 views

Show that $GL_n(\mathbb{Z})\ncong SL_n(\mathbb{Z})$ for $n \geq 3$?

Given that ${SL}_n(\mathbb{Z})$ is generated by the set of transvection matrices. Show that ${GL}_n(\mathbb{Z})\ncong {SL}_n(\mathbb{Z})$ for $n \geq 3$. So the set of transvection matrices ...
2
votes
1answer
56 views

Find a normal matrix with a given characteristic polynomial

Find a normal matrix with characteristic polynomial $t^2 + 4$ and eigenspace $E_{2i} = span {(1 \ 3i)^t}$ Since any vector in different eignespaces are perpendicular to each other, so i compute ...
2
votes
4answers
48 views

Image of linear system

I found the determinant and null space of the matrix in previous exercises, but I am having trouble understanding how to find the image of following matrix. "Given the matrix $$A_{a} = ...
2
votes
1answer
40 views

Is there any way to make matrix multiplication 'act' commutative?

I've come across a certain matrix equation that I'm trying to manipulate and solve; made more difficult because the equation is arbitrarily long and interspersed with matrix powers. The matrix in ...
2
votes
2answers
53 views

Evaluate product of trace of two matrices

I have a question. Let A be a positive semi-definite matrix, H be a positive definite matrix. Is the following inequality right: $Tr(AH).Tr(AH^{-1}) \geq (Tr(A))^2$? I tried to take some concrete ...
2
votes
2answers
46 views

Compute the indicated power of a matrix

Compute the indicated power of the matrix: $A^8$ $ A = \begin{bmatrix}2&1&2\\2&1&2\\2&1&2\end{bmatrix} $ I calculated the eigenvalues: $ \lambda_1 = \lambda_2 = 0, \lambda_3 ...
2
votes
1answer
67 views

An example of a sentence $\sigma$ s.t. $\text{GL}_n(\mathbb{Q}(\sqrt{3})) \models \sigma$ and $\text{GL}_n(\mathbb{Q}(\sqrt{2})) \not\models \sigma$

A. I. Mal'cev proved the following remarkable result concerning the elementary equivalence of general linear groups: given fields $\mathbb{F}_{1}$ and $\mathbb{F}_{2}$ and natural numbers $m$ and $n ...
2
votes
1answer
41 views

Positive Definite or Negative Definite Matrix

Say I have a matrix A and I'm trying to determine for which values of $a$ makes the matrix positive definite and which values make it negative definite. $$A = \begin{bmatrix} 2+a &2 & \sqrt ...
2
votes
1answer
75 views

How to compute the center of $SU(2)$?

It is stated in our lecture notes without proof that the center of $SU(2)=\{\pm 1\}$. I understand how to find the center of $SO(3)$, which is $\{1\}$ and that is given in the notes, is that somehow ...
2
votes
1answer
140 views

Second derivative of a composition?

Let $g:\mathbb{R}^n\to\mathbb{R}^p$, $f:\mathbb{R}^p\to\mathbb{R}$ and define the composition $h(x) = f(g(x))$. The gradient of $h$ with respect to $x$, $\nabla h\in\mathbb{R}^{1\times n}$, is given ...
2
votes
2answers
67 views

Proving that $(A^n)^{-1} = (A^{-1})^n$ for invertible matrix $A$.

I have seen a proof of the fact that for an invertible matrix $A$, $A^n$ is also invertible and $$ (A^n)^{-1} = (A^{-1})^n. $$ The proof was by induction and it was mentioned that one has to use ...
2
votes
1answer
31 views

is there a Relationship between duplicity of EigenValue and dimension of it's EigenSpace?

giving characteristic polynomial of a matrix (Which has eigenvalues with it's duplicity) how can we understand the dimension of eigenspace of each eigenvalue without direct calculation? in addition, ...
2
votes
1answer
100 views

Matrix induction proof

Given the following $\lambda_{1}=\frac{1-\sqrt{5}}{2}$ and $\lambda_{2}=\frac{1+\sqrt{5}}{2}$ How do I prove this using induction: $\begin{align*} A^k=\frac{1}{\sqrt{5}}\left(\begin{array}{cc} ...
2
votes
2answers
47 views

Suppose $\Vert Ax\Vert _{2}=\Vert Bx\Vert _{2}$ for all $x\in\mathbb{R}^{n}$ , does that imply $A=B$ or $A=-B$?

Suppose $A,B\in\mathbb{R}^{n\times n}$ are matrices such that $\Vert Ax\Vert _{2}=\Vert Bx\Vert _{2}$ for all $x\in\mathbb{R}^{n}$ , does that imply $A=B$ or $A=-B$. I couldn't come up with a ...