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
votes
1answer
38 views

Dimension of Vector Spaces

Can anybody help me finding out the dimension of the vector spaces: A: A is $n\times n$ real upper triangular matrices. A: A is $n\times n$ real symmetric matrices. A: A is $m\times n$ real ...
2
votes
2answers
47 views

Geometrical Interpretation of Matrix Multiplication

I am stuck up with this question from my Linear Algebra Assignment which states to explain geometrically why $$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix}-1 & 0 \\ 0 ...
1
vote
1answer
64 views

Cokernel as direct sum of cyclic groups

I am asked to reduce the matrix $ \left( \begin{array}{ccc} 3 & 1 & -4 \\ 2 & -3 & 1 \\ -4 & 6 & -2 \end{array} \right)$ to diagonal form over $\mathbb{Z}$ and then write the ...
0
votes
3answers
129 views

Some questions about notation in “$[T]_\alpha^\beta$”

I just have a few questions about the general meaning of the notation "$[T]_\alpha^\beta$". I would really appreciate if someone would dumb it WAY down to the most basic level (no assumptions, no ...
2
votes
1answer
39 views

Confusion with the notation $L_A$

My linear algebra class went from 0-100 real quick. I've attended every single lecture (so I know I haven't missed out on anything); however, very recently he has been using the notation $L_A$ for a ...
0
votes
1answer
80 views

Find the dimension of the set $S=\{B_{6\times 6}: AB=BA\}$.

Let , $A_{6\times 6}$ diagonal matrix with characteristic polynomial $x(x+1)^2(x-1)^3$. Find the dimension of the set $S=\{B_{6\times 6}: AB=BA\}$. From characteristic polynomial of $A$ , first ...
1
vote
1answer
69 views

Morphism of $k$-algebras between matrix rings over $k$

Let $k$ be a field and $f:M_n(k) \to M_m(k)$ be a morphism of $k$-algebras ($n,m \in \mathbb N$). Prove that $n$ divides $m$. I have no idea what to do here. I thought that maybe I should take an ...
0
votes
2answers
148 views

Show that V is a vector space over the set of real numbers when V is the set of all real 3x3 matrices

Wondering how one would go on about this. V is the set of all real 3 × 3 matrices. How can it be shown that V is a vector space over the set of real numbers and what would be the dimension of and ...
0
votes
1answer
26 views

How do you find the matrix relative to a basis?

I'm having trouble knowing where to start. I've been given the problem: Let $\ B = \{1, x, sin(x), cos(x)\}$ be a basis for a subspace $\ W$ of the space of continuous functions, and let $\ Dx $ be ...
2
votes
2answers
98 views

What is the difference between $n$-tuples, $m \times 1$ and $1 \times n$ matrices?

Isn't the tuple different structure from $m \times 1$ or $1 \times n$ matrix? Why can you mix them?
4
votes
3answers
2k views

Skew-symmetric matrix subspace dimension and basis

If $M$ is the vector space of $2\times 2$ real matrices, then I can show that $$ \{A \in M \mid A^\mathrm{T}=-A \} $$ is a subspace of $M$, since $$ \left[ \begin{array}{cc} x & z \\ -z & ...
0
votes
0answers
32 views

Show: If $v \in E^{\perp}$ then it can be written as $v=w+c_1w_1+c_2w_2+\dots +c_kw_k$

(i) Assume that $B = \{w_1,\dots,w_k\}$ is an orthogonal basis for $E$. Let $v \in E^{\perp}$ such that $v\neq O_{V}$. Prove that $v=w+c_1w_1+c_2w_2+\dots +c_kw_k$ for some nonzero $w\notin E$ and ...
1
vote
1answer
69 views

Find the Jordan form of a 4 x 4 matrix

Find the Jordan Form of $$ A=\left[\begin{array}{cccc} 0 & -16 & 0 & 0\\ 1 & 8 & 0 & 0\\ 0 & 0 & 0 & -6\\ 0 & 0 & 1 &5 \end{array}\right] $$ ...
6
votes
2answers
139 views

Finding minimum from matrix

Consider following $3\times 3$ matrix. $\begin{pmatrix}3&6&9\\ 2& 4 &8\\ 1 &5& 7 \end{pmatrix}$ I need to find combination of three numbers where each number ...
1
vote
2answers
37 views

find the smallest interval in which the eigen value of the matrix lie

$$ \begin{bmatrix} 3 & 2 & 2 \\ 2 & 5 & 2 \\ 2 & 2 & 3 \\ \end{bmatrix} $$ I was practicing questions on Matrices & Determinants ...
0
votes
1answer
262 views

Find the coordinate matrix of a polynomial with respect to a non-standard basis

I'm stuck on this question here: Find the coordinate matrix of $2-4x-3x^2$ with respect to $B = {2, x^2-1, 1-2x-x^2}$ I did the following: $a(2) + b(x^2 - 1) + c(1-2x-x^2) = 2-4x-3x^2$ But now I'm ...
1
vote
1answer
39 views

What seems to be the minors of the Adjugate matrix $\text{adj}(A)$ of a square matrix $A$?

It is by definition that entries of the adjugate matrix $\text{adj}(A)$ are the corresponding $(n-1)$-minors of $A$ (up to a sign). What can we say about the $k$-minor of $\text{adj}(A)$ in relation ...
0
votes
6answers
3k views

Why does a matrix have determinant zero if one row is the sum of two other rows?

So basically here I am trying to understand why it is like that? Suppose Matrix $$ A = \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ a+d & b+e & c+f \end{array} \right) ...
1
vote
1answer
62 views

Finding matrix $A$ knowing that $A^2 = B$

Let $B$ be the $3\times3$ matrix $$ \begin{pmatrix} 1&8&5\\ 0&9&5\\ 0&0&4 \end{pmatrix}. $$ How can I find a triangular matrix $A$ with positive diagonal entries such that ...
0
votes
2answers
75 views

if $rank{(A - \lambda I)^k} = rank{(B - \lambda I)^k}$ then $A$ is similar $B$

Let $A,B \in M_n(\mathbb{R}).$ Suppose for all $\lambda \in \sigma (A)$ and for all $k \geq 0,$ we have $\mathrm{rank}(A - \lambda I)^k = \mathrm{rank}(B - \lambda I)^k.$ Then why are $A$ and $B $ ...
0
votes
1answer
61 views

If each eigenvalueof $A$ is either $+1$ or $-1$ $ \Rightarrow$ $A$ is similar to ${A^{ - 1}}$

Let $A \in {M_n}$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$?
1
vote
1answer
60 views

Jordan form of different matrices

Suppose you have a 4x4 matrix with the characteristic polynomial equal to the minimal polynomial $m_F(x)=C_F(x)=(x-3)^2(x+2)^2$. Find the Jordan form. Is this the correct solution? $$ ...
1
vote
1answer
31 views

A question about non-linear least square method…

I am trying to fit a set of points into a sine function, using nonlinear least square method. The final step to obtain the derivative of its parameters is given by the equation (8) of: ...
0
votes
1answer
47 views

How can we find if a matrix is full column rank

If $A$ is an $n*k$ matrix with complicated form of elements. How can I show this matrix is full column rank? By complicated form I mean there is no known form for the elements of $A$.
1
vote
2answers
64 views

How is it distinguished in matrix multiplication which is the vector and which is the matrix representing a linear transformation?

The terminology that is used everywhere when applying a matrix to a "vector" is considered is this: the matrix represents a linear transformation and there is a row or column vector. But a matrix can ...
0
votes
1answer
84 views

Reverse Order Laws of M-P pseudoinverse

When I was writing a literature survey on Moore-Penrose pseudoinverse (literatures like this one, and this one), I encountered with the following equality which was named as reverse order law: ...
2
votes
0answers
16 views

Cartan matrices: motivation and intuitive examples?

could anyone provide me with a sketch of the motivation that gave rise to Cartan matrices in abstract (homological) algebra, Lie algebrae and so on? Which was the trigger or the need for them? It ...
26
votes
9answers
2k views

What does it mean to represent a number in term of a $2\times2$ matrix?

Today my friend showed me that the imaginary number can be represented in term of a matrix $$i = \pmatrix{0&-1\\1&0}$$ This was very very confusing for me because I have never thought of it ...
0
votes
1answer
74 views

norms of row matrices

Let $x_1,\ldots,x_m,y_1,\ldots,y_m$ be $k\times k$ sqaure matrices and assume $\|x_j\|\leqslant\|y_j\|$ for all $j=1,\ldots,m$ (the norm in $B(\ell_2^k)$). Now define the block matrices $x,y\in ...
2
votes
2answers
1k views

Derivative with respect to a matrix

How do we start with the matrix differentiation of this kind of equation? $$ V = \big[ y_t - Cx_t \big]^T R^{-1} \big[y_t - Cx_t \big] $$ here $x_t$ and $y_t$ are vectors and $C$ and $R$ are ...
-1
votes
1answer
30 views

Is this true that $A$ and $A^*$ are unitary equivalent(or equivalent)? .

Let $A \in {M_n}$.Is this true that $A$ and $A^*$ are unitary equivalent(or equivalent)?
0
votes
1answer
56 views

If ${\sum\limits_{i = 1}^n {\left| {{\lambda _i}} \right|} ^2} = \sum\limits_{i = 1}^n {{\sigma _i}^2} \Rightarrow$A is normal matrix

Let $A \in {M_n}$ have eigenvalues ${\lambda _1}.....{\lambda _n}$ and singular values ${\sigma _1}.....{\sigma _n}$ and suppose ${\sum\limits_{i = 1}^n {\left| {{\lambda _i}} \right|} ^2} = ...
1
vote
1answer
29 views

finding matrix represention for linear transformation for field extension

need some clarification. given an extension field K over F with F-linear transformation, for $\alpha \in K$, $f_\alpha(k) = \alpha \cdot k$ i.e. multiplication on the left. I need to find the ...
2
votes
0answers
39 views

Constrained zero diagonal low rank approximation of a matrix with zero diagonal

Suppose that you have a $n\times n$ matrix $A$ that is symmetric and has zero diagonal, such as for example $$ A=\pmatrix{ 0 & 2 & 2\\ 2 & 0 & 1\\ 2 & 1 & 0}, $$ and you want ...
0
votes
2answers
28 views

proof a theorem in linear algebra

prove that if λ1 and λ2 are two distinct eigenvalues of a matrix A and λ1 , λ2 are corresponding eigenvectors, respectively, then α1 and α2 are linearly independent please help... thank you...
1
vote
0answers
49 views

Formula for powering a matrix not working for all matrices

I'm currently learning about matrices and was asked to show that this formula works for powers of $M$. $$M^n = nM-(n-1)I$$ Where $M$ is the matrix (show below), $n$ is the exponent an $I$ is the ...
5
votes
3answers
58 views

If $A$ is a square matrix and $Ax = b$ has a unique solution for some $b$, is $A$ necessarily invertible?

Let $A$ be a square matrix. Suppose that $A x = b$ has a unique solution for some $b$. Is $A$ necessarily invertible? I said no because the invertible matrix theorem states that $A x = b$ has a ...
1
vote
2answers
73 views

Why does $tr({A^*}A) = \sum\limits_{i = 1}^n {{\sigma _i}^2} $?

Let $A \in {M_n}$ have eigenvalues ${\lambda _1}.....{\lambda _n}$ and singular values ${\sigma _1}.....{\sigma _n}$. Why does $tr({A^*}A) = \sum\limits_{i = 1}^n {{\sigma _i}^2} $?
1
vote
1answer
58 views

Distribution of a matrix product $\mathbf{a}^{H}\mathbf{H}\mathbf{b}$

Could someone help prove the following: I have two independent random vectors $\mathbf{a} \in \mathbb{C}^{M \times 1}$ and $\mathbf{b}\in \mathbb{C}^{N \times 1}$. Both $\mathbf{a}$ and $\mathbf{b}$ ...
-1
votes
1answer
101 views

Image of the product of a Matrix and its transpose

If $A$ is an $n\times m$ matrix, is it necessarily true that $\text{im}(A)=\text{im}(AA^T)$ where $A^T$ is the transpose of $A$.
1
vote
0answers
20 views

Similarity of orthogonal matrices

Prove that for any $M$ in $SO(3)$, there is a matrix $P$ in $SO(3)$ and a real $\alpha$ such that $$PMP^{-1} = \left[ \begin{matrix} \cos\alpha & \sin\alpha & 0 \\ ...
0
votes
1answer
87 views

Eigen vectors for matrix with unknown constants?

I have the following matrix: $$\begin{bmatrix}\alpha&0&0\\\beta-\alpha&\beta&0\\1-\beta&1-\beta&1\end{bmatrix}$$ So far I have worked out the polynomial to be: ...
-1
votes
1answer
27 views

How to find the matrix of the transformation relative to the basis?

Let $T:P_2\to P_2$ be the linear operator defined by $$T(a+bx+cx^2)=(3a+2b+4c)+(2a+2c)x+(4a+2b+3c)x^2$$ Find the matrix of the transformation $T$ relative to the basis $B=\{1,x,x^2\}$.
3
votes
1answer
66 views

Comparing/contrasting hyperbolic and Euclidean geometry - or, on how ${\rm PSO}_2(\Bbb R)$ sits inside ${\rm PSL}_2(\Bbb R)$

I am studying hyperbolic geometry, in particular comparing and contrasting it with familiar Euclidean geometry. Let $\Bbb E$ be the Euclidean plane, and $G={\rm Iso}^+(\Bbb E)$ be the group of ...
3
votes
2answers
90 views

Show a matrix satisfying $A^2 − 8A + 15I = 0$ is diagonalisable.

A square matrix $A$ (of some size $n × n$) satisfies the condition $A^2 − 8A + 15I = 0$. Show that this matrix is similar to a diagonal matrix. I know that we must show that 5 and 3 are the ...
4
votes
1answer
64 views

Question about eigenvalue of Hermitian matrix

This is an eigenvalue problem I found. Let $A$ be an $n$-by-$n$ Hermitian complex matrix and $u$ is a vector in $C^n$ such that $u^*u=1$. Let $k=u^*Au$. Show that there exists an eigenvalue $r$ of ...
1
vote
3answers
44 views

How can I show that the dimension of span is . . .?

The Krylov subspace generated by $n$-by-$n$ matrix $A$ is defined by : $K_k(A,x)=span\{x,Ax,A^2x,...,A^{k-1}x\}$ How do I show its dimension is at most $k$? I only know that $dim(span (V))=rank ...
2
votes
1answer
40 views

Proving matrix exponent property [closed]

How can I prove the following equation. I have tried but i couldn't. $$\exp(A(t_2+t_1))=\exp(At_2)\cdot \exp(At_1)$$ $A$ is a matrix Will I use state-transition matrix or what ? Thank you...
0
votes
2answers
94 views

sum of two wishart matrices

Assuming $\mathbf{H}_1\in\mathbb{C}^{K\times M}(M>K),\mathbf{H}_2\in\mathbb{C}^{K\times M} $, the entries of $\mathbf{H}_1\text{ and }\mathbf{H}_2$ are all i.i.d. $\mathcal{CN}(0,1)$. I know that ...
0
votes
1answer
46 views

How to see that $A = A^{-1}$ and $A^2 = A$ as quick and easy as possible without computer aid

I'm wondering which are the quickest/easiest methods to identify that the following relations hold for any given matrix: $A = A^{-1}$ and $A^2 = A$ On a computer it's easy and quick to identify if ...