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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3
votes
3answers
29 views

Exponential of a matrix with elements $\cos t \& \sin t$

I want to calculate $e^{A}$ of the matrix $A$: $$\left ( \begin{array}{cc} \cos t & \sin t \\ -\sin t & \cos t \end{array} \right )$$ I tried to use $e^{At}=P\ \mbox{diag}(e^{\lambda t}) ...
1
vote
1answer
17 views

Find the Matrix of T with respect to basis B.

A linear transformation $T : P_2 \rightarrow P_2$ is given by $T(a+bx+cx^2) = (a−b+c)+(b+c)x+(2b−a)x^2$. It is given that the set $B=\{1+x+x^2, x+x^2, x^2\}$ is a basis for $P_2$. (a) Find the matrix ...
0
votes
0answers
15 views

Eigenvalues of the product of these 2 matrices

I am currently working on the following problem: $$ \textbf{Y} \in \mathbb{R}^{n \times q}, \textbf{X} \in \mathbb{R}^{n \times p} $$ $$ \textbf{Q} \in \mathbb{R}^{n \times n} (\hbox{symmetric, ...
-4
votes
1answer
33 views

How can we find the inverse matrix? [on hold]

How can we find the inverse of the matrix $$K=\begin{pmatrix}-(x+y) & y & 0 \\ y & -(y+z+w) & w\\ 0 & w & -(w+f)\end{pmatrix}$$ ??
2
votes
1answer
28 views

Inequality in matrix norm

Let $\|\cdot\|$ be matrix norm on $M_n$.Why does $\|A\|_2 \le \|A\|^{\frac{1}{2}} \|A^*\|^{\frac{1}{2}}$? ($\|A\|_2 = \displaystyle\max_{\|x\|_2 = 1} \|Ax\|_2$)
3
votes
3answers
25 views

How to prove that this matrix is positive definite?

Let $\mathbf{A}=\begin{pmatrix}a^2+b^2 & b^2 & b^2 & ... & b^2 \\ b^2 & a^2+b^2 & b^2 & ... & b^2\\ \vdots & b^2 & \ddots & & b^2 \\ b^2 & \dots ...
6
votes
3answers
73 views

Commutative property of matrix multiplication (or lack thereof)

Assuming $A$ and $B$ are invertible matrices and are of proper dimensions to be multiplied (say, $2\times2$), is the following expression correct for all examples of matrices $A$ and $B$? ...
5
votes
1answer
44 views

Why does ${\lambda _i}(A) \ge {\lambda _i}(B)$?

Let $A,B \in {M_n}$ are Hermitian and $A-B$ has only nonnegative eigenvalues.Why does ${\lambda _i}(A) \ge {\lambda _i}(B)$ (for $i=1,2,\ldots,n$) ?
3
votes
1answer
21 views

Clarification regarding linear transformation from $\mathbb{R}^{2\times 2}$ to $\mathbb{R}^{2\times 2}$

Every linear transformation has a matrix representation, once we've chosen a basis. Suppose I define a linear transformation $T:\mathbb{R}^3\rightarrow \mathbb{R}^2$ as follows: $T(x,y,z)=(x+y,y+z)$. ...
0
votes
2answers
60 views

Matrices $\begin{pmatrix} a &b \\ c &d \end{pmatrix}\begin{pmatrix} x\\y \end{pmatrix}=k\begin{pmatrix}x\\y \end{pmatrix}$

If $\begin{pmatrix} a &b \\ c &d \end{pmatrix}\begin{pmatrix} x\\y \end{pmatrix}=k\begin{pmatrix}x\\y \end{pmatrix}$, prove that $k$ satisfies the equation $k^2-(a+d)k+(ad-bc)=0$. If the ...
1
vote
2answers
27 views

Do $J$, the all-ones matrix of even order, always have eigenvectors consisting of entries $-1, 1$ only?

Do $J$, the all-ones matrix of even order, always have eigenvectors consisting of entries $-1, 1$ only? It seems so, vector having all its entries $1$ is one eigenvector for larest eigenvalue $n$ ...
3
votes
2answers
76 views

How many orthogonal matrices are there

this might sound like a stupid question, but what I mean is: You need $n \times n$ elements to define a square matrix $\in R^{n \times n}$. How many element do I need to define an orthogonal matrix? I ...
2
votes
0answers
31 views

A question on a matrix identity

Sorry for the not very specific title. I was hoping I could get some help with a result I do not understand. The following is from a book I am reading. What I do not understand is how from 9.9.6 one ...
0
votes
0answers
18 views

Relation between Tensor-hom adjunction and adjugate matrix

Let $R\to S$ be a ring homomorphism, let $M,N$ be $S$-modules and $Q$ an $R$-module. Then, we have $$\textrm{Hom}_R(M\otimes_S N,Q) \cong \textrm{Hom}_S(M,\textrm{Hom}_R(N,Q).$$ I want to know ...
-1
votes
0answers
23 views

How to determine the matrix $A$, which belongs to mapping in the standard bases $\{1, t, t^2\}$ [on hold]

Linear mapping $A$ that picture from the space of real polynomials of degree $2$ in itself is under the rules $(Ap)(t) := (tp(t))' + 6t \int p(x)\,dx$. Specify a similar diagonal matrix and an ...
-1
votes
1answer
27 views

The inverse of a matrix in which the sum of each row is $1$

Let $A$ be an invertible 10x10 matrix with real entries such that the sum of each row is $1$. Then choose the correct option. The sum of the entries of each row of the inverse of $A$ is $1$. The sum ...
4
votes
2answers
28 views

An invertible sparse matrix?

I'm not entirely certain about how to tackle this problem.... I hope you ladies and gents can help :) If $M\in M_{n\times n}(\mathbb{R})$ be such that every row has precisely tow non-zero entries, ...
1
vote
1answer
43 views

Show that there do not exist nonsingular matrices $P,Q ∈ M_{n×n}(F )$ satisfying $PAQ = A^T$ for all $A ∈M_{n×n}(F )$.

Let $F$ be a field and let $n$ be a positive integer. Show that there do not exist nonsingular matrices $P,Q ∈ M_{n×n}(F )$ satisfying $PAQ = A^T$ for all $A ∈M_{n×n}(F)$. (Exercise 438 from ...
0
votes
0answers
30 views

what are the principal applications of symmetric matrices in physics giving examples of it's applications? [on hold]

symmetric matrix is useful in many areas of sciences such as : physics . i'm very interested to know the suspect and some interesting applications of " Symmetric" matrix in physics or any branch ...
0
votes
0answers
37 views

Prove a matrix is non-negative. [on hold]

Let $\textbf{r}_1$ and $\textbf{r}_2$ be $n \times n$ symmetric, diagonally dominant, Metzler matrix with eigenvalue $\max(|\lambda_i|)<1$ for both $\textbf{r}_1$ and $\textbf{r}_2$. Let ...
-1
votes
1answer
24 views

QR and Cholesky decomposition

A while ago I asked for help to develop a polynomial regression model using least squares, where the system was solved by cholesky decomposition, you can check it here Cholesky Polynomial Regression ...
1
vote
0answers
16 views

Why covariance constraint subsumes the average power constraint?

I am studying an optimization problem in the form of \begin{equation} \begin{aligned} &\underset{p(x)}{\text{maximize}} & & W\\ & \text{subject to} & & 0 \preceq K_{X} ...
0
votes
4answers
42 views

How can I prove that this matrices statement is false?

How can I prove that this is not true: If for matrices A, B and C, AB=AC and A is not the zeroth matrix, then B=C.
0
votes
0answers
30 views

Orthogonal projection af a $5\times3$ matrix onto a subspace spanned by two of its vectors.

As a part of a data analysis exercises I need to project a matrix that contains $5$ observations of $3$ variables onto a plane spanned by two of those variables. I can't really imagine this. What is ...
3
votes
3answers
62 views

If $A$ is a matrix, and $A^2=I$, then can I say that $|A|= \pm1$?

$A^2=I$ Take determinant on both sides: $$|A^2|= |I| $$ $$|A|^2= 1$$ $$|A| = +1 \text{ or } -1$$ Is this proof correct?
0
votes
2answers
17 views

how to prove the equivalent statements in matrix?

Here is the equivalent statements: (a) A is invertible (b) Ax=0 has only the trivial solution. (c) The reduced row echelon form of A is I (d) A is expressible as a product of elementary matrices. (e) ...
-1
votes
0answers
18 views

How to find the transformation P in the standard base? [on hold]

Let be $P: R^3 \to R^3$ projector to the plane $x + y - z = 0$ along the line $x = y = z.$ Show that the P* is also a projector. Where is projected and along what? In $R^3$ we have standard scalar ...
0
votes
0answers
12 views

Cholesky factorization for positive semidefinite matrices

I know that a matrix $A$ is positive definite and symmetric if and only if there exists a lower triangular matrix $L$ with nonzero diagonal such that $A = LL^T$. I'm wondering if it similarly holds ...
0
votes
0answers
18 views

Behavior of eigenvalues of certain matrices

I am trying to analyze the behavior of the 2 highest eigenvalues of matrices of this form : Symmetric $n*n$ matrices that contains only : $1/k$ (for fixed k), -1,1 and 0. My hope is to find some ...
0
votes
2answers
41 views

If I have a matrix M=[A,B;0,C], how do I prove that rank(A)+rank(C)<=rank(M)?

. . . . . . . A . . B . . . . . . . 0 0 0 . . . 0 . 0 . C . 0 0 0 . . . If I have a matrix $M$ as displayed in the text above ($A$ ...
1
vote
0answers
43 views

Is there a closed form expression for $(A^T\Sigma A)^{-1}$ when $A$ is not square?

I need to find the inverse of the matrix $A^T\Sigma A$. Matrix $A$ has dimensions $5\times 2$. Matrix $\Sigma$ has dimensions $5\times 5$, and it is symmetric and positive-definite. I need to ...
3
votes
1answer
60 views

$A,B$ be Hermitian.Is this true that $tr[(AB)^2]\le tr(A^2B^2)$?

Suppose $A,B \in {M_n}$ be Hermitian.Is this true that $tr[(AB)^2]\le tr(A^2B^2)$?
1
vote
0answers
26 views

Equivalence class of matrices on linear form

We well know that if $M$ is a matrix on a field $k$ then the equivalence class of $M$ is uniquely determined by its rank (where $A \sim B$ if $\exists P,Q $ invertibles such that $PAQ^{-1}=B$). ...
1
vote
0answers
30 views

Why does $rank(A) \ge \dfrac{{{{(trA)}^2}}}{{(tr{A^2})}}$? [duplicate]

Let $A \in {M_n}$ and Hermitian.Why does $rank(A) \ge \dfrac{{{{(trA)}^2}}}{{(tr{A^2})}}$?
0
votes
3answers
69 views

Let $A = \left( {\begin{array}{*{20}{c}} 0&1\\ 0&0 \end{array}} \right)$.What is numerical range $A$ [on hold]

Let $A = \left( {\begin{array}{*{20}{c}} 0&1\\ 0&0 \end{array}} \right)$. How to find its numerical range $W(A) = \{ {x^*}Ax:x \in {S^1}\}$?
2
votes
0answers
26 views

Closed form of a matrix product

Is there any closed form or a bound for a matrix product of this kind $$ P=\prod_{i=1}^n \begin{pmatrix} 1-a & a \\ b_i & 1-b_i \end{pmatrix}, \quad a,b_i \in [0,1] $$ for an arbitrary ...
1
vote
0answers
23 views

Comprehensive easy to understand resource for learning matrix decompositions?

I am working on my thesis which is widely depended on knowledge about matrix decompositions. I have studied linear algebra with the help of YouTube videos, MITOpenCourseWare videos and Prof.Gilbert ...
0
votes
0answers
31 views

How to get real irreducible matrix representations from the complex irreducible matrix representations?

I'm trying to get real symmetry adapted orbitals for molecules with icosahedric symmetry (point groups $I$ and $I_h$) using the complete projector operator (truly projector if i=j): \begin{equation} ...
-5
votes
0answers
29 views

augmented matrix question [on hold]

Please show me the augmented matrix solution with steps for the system $$ \begin{cases} 3x + y+z=18 \\ 4x + 2y+3z=12 \\ 7x + 8y+5z=9 \end{cases} $$
0
votes
1answer
15 views

Finding an ordered basis to diagonalize Transpose matrix.

We define $T : M_{n \times n}R \to M_{n\times n}R$ by $T(A) = A^t$. We can write the matrix representation of this transformation as: $[T]_\beta^\beta = \begin{pmatrix} ...
1
vote
3answers
31 views

For which values of $a$ does the matrix can be diagnolized?

Given $$A=\begin{pmatrix} 2 & 0 & 0\\ a & 2& 0\\ a+3 & a &-1 \end{pmatrix}$$ For which values of $a$ can $A$ be diagonal? I found that $p_A(x)=(x-2)^2(x+1)$ and tried to ...
-1
votes
0answers
15 views

Which matrix A belongs to a transformation A* in the standard base (1, t, t^2) of the real polynomial space P2 (R)? [on hold]

Linear transformation is given by $%![transformation ][1]$ $$ (Ap)(t) := tp'(t) + \int_0^t xp''(x)\,dx $$ and a scalar product $%![scalar][2]$ $$ (p,q) := p(-1)q(-1) + p(0)q(0) + p(1)q(1) $$ Which ...
0
votes
1answer
24 views

What is $\max(\operatorname{Re} \{ \frac{x^* Ax}{x^* x}:0 \ne x \in C^n\} )$?

Let $A = \left( \begin{array}{*{20}{c}} 1&2\\ 0&1 \end{array} \right)$. What is $\max\left(\operatorname{Re} \left\{ \dfrac{x^* Ax}{x^* x}:0 \ne x \in C^n\right\} \right)$?
5
votes
1answer
41 views

Understanding a part of a proof involving Hilbert-Schmidt norm

I came across a proof I do not seem to understand fully, a screenshot is provided below. my concerns are the following: Why does the fact that $||T||_2 = ||UT||_2$ for every unitary U, allow us ...
1
vote
1answer
32 views

What are the facts used in each step of this proof?

What are the facts used in each step of this proof ? Suppose that $A\in F^{nm}$ and $B\in F^{ml}$ $$\begin{align}rank A + rank B &= rank\begin{bmatrix}0 & A\\B & 0\\ \end{bmatrix}\\ ...
2
votes
5answers
35 views

Eigenvalues of different symmetric $(2n+1)\times(2n+1)$ matrix

I ve looked at other similar post but I could not find help with them
0
votes
0answers
15 views

find Jordan form

Determine the jordan form of $A = \begin{pmatrix} 1 & 2 & 3\\ 0 & 4 & 5\\ 0 & 0 & 4 \end{pmatrix} $ First, I find the characteristic polynomial. $C_A(x)=(x-1)(x-4)^2$. ...
1
vote
4answers
30 views

Does an arbitrary matrix $X \in M_{n \times p}$ have a SVD?

I have proven, as below, that if $X \in M_{n \times n}$ is symmetric, then it has a SVD. $D(\lambda_i) = \text{Diag}(\lambda_i)$ is a diagonal matrix with entries $\lambda_1, \lambda_2, \dots$. ...
-2
votes
0answers
21 views

Is this proof correct? Matrix ring. Center. [duplicate]

I found this https://crazyproject.wordpress.com/2010/08/23/the-center-of-a-matrix-ring-over-a-commutative-ring-is-precisely-the-scalar-matrices/ and I have very bad day, so I ask you to confirm.
0
votes
1answer
39 views

Prove that if $A,B\in M_n(\mathbb{F})$ are $(n-1)$-nilpotent then they are similar.

If $A,B\in M_n(\mathbb{F})$ are $n-1$ nilpotent, prove they are similar. Can I say that, since their minimal polynomial is $X^{n-1}$ they are similar? I know that If $A,B$ are similar, they have ...