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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5
votes
2answers
61 views

Maximal order of elements of $\textrm{SL}(n, \mathbb{Z})$

In the case of $\textrm{SL}(2, \mathbb{Z})$, I know that the order of any finite order matrix in this group is at most $6$. This follows from the fact that $\textrm{SL}(2, \mathbb{Z}) \cong ...
0
votes
2answers
71 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
vote
2answers
35 views

$\frac{1}{n} \left( \sum_{i=1}^n a_{ii} \right)^2 \le \|(a_{i,j})\|^2_F$

I want to show that one can estimate the Trace of a matrix by the Frobenius norm of a matrix. $$ \frac{1}{n}\left( \sum_{i=1}^n a_{ii} \right)^2 \le \|(a_{i,j})\|^2_F.$$ Unfortunately, I think that ...
0
votes
0answers
75 views

Trace of a product of 2 matrices

I have the following problem. Let $\textbf{Y} \in \mathbb{R}^{n \times q} \; n>q, \textbf{H} \in \mathbb{R}^{n \times n}$ such that $\textbf{H}$ is idempotent ($\textbf{H}^{2} = \textbf{H}$) and ...
0
votes
1answer
41 views

When not to use rref for finding eigenvectors?

I have this matrix which is corresponding to an eigenvalue $\lambda = 50$: \begin{array}{ccc|c} 27 & 0 & 36&0\\ 0&0&0&0 \\ 36 & 0& -27&0 \end{array} The ...
1
vote
2answers
49 views

Solution of the differential equation?

First of all, I am sorry that I could not pick up a better title for the question :) Today, I am working with DE questions involving symmetric positive definite matrices, and this is the second ...
0
votes
1answer
131 views

Solving a multivariate non-linear system of equations using Newton's method

I'm doing a report on the mathematics of GPS, and I have the following equations to solve for $x$, $y$, $z$ and $t_b$: I'm using Newton's method, with the Jacobian matrix of partial derivatives ...
0
votes
1answer
32 views

A question about matrix norm

We know that${\left\| {\left| A \right|} \right\|_2} = \mathop {\max {{\left\| {Ax} \right\|}_2}}\limits_{{{\left\| x \right\|}_2} = 1} $.Let $A$ is Hermitian and $A \in {M_n}(C)$.Is this true that ...
1
vote
2answers
256 views

“Averaging” transformation matrices?

I have a question on how best to "average" transformation matrices. Say that I have n number of 4x4 transformation matrices, and I wanted to find a matrix that approximated each one of the n 4x4 ...
1
vote
2answers
259 views

If a vector v is an eigenvector of both matrices A and B, is V an eigenvector of A+B? [closed]

If so, is there a proof for this? I have been stuck trying to validate the statement and would love some insight.
4
votes
2answers
165 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 ...
3
votes
1answer
43 views

Linear combination of matrices over finite and infinite fields

Let $F\subset K$ be the fields. Let $A_1,\ldots, A_m$ be the $n\times n$ matrices over the field $F$, and $c_1,\ldots,c_m\in K$ such that $c_1A_1+\cdots+c_mA_m$ is invertible. How to prove that for ...
6
votes
2answers
60 views

Quotient group $\mathbb Z^n/\ \text{im}(A)$

Let $A$ be an $n \times n$ matrix with integer coefficients and nonzero determinant. Can we say something about $ \mathbb{Z}^n /\ \text{im}( \phi )$ (here $\phi : v \mapsto Av$ )? This problem ...
0
votes
2answers
97 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} ...
6
votes
3answers
301 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 $B$? ...
2
votes
0answers
46 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 ...
-1
votes
1answer
102 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 ...
2
votes
1answer
52 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 ...
3
votes
3answers
75 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
416 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 ...
5
votes
1answer
60 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 $i=1,2,\ldots,n$) ?
2
votes
2answers
94 views

Derivatives of component inverse functions

I might have missed the point of the following questions. Anyone kindly give a suggestion? Let $f:\mathbb{R}_\mathbf{x}^3\to\mathbb{R}_\mathbf{y}^3$ and ...
3
votes
3answers
48 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 ...
3
votes
1answer
25 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
65 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 ...
2
votes
2answers
56 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 $n$ ...
2
votes
3answers
79 views

Find an arbitrary power of a lower triangular matrix of size $3\times 3$

Let $F$ be a field and let $A=\begin{bmatrix}a&0&0\\1&a&0\\0&1&a\end{bmatrix}\in\mathscr{M}_{3\times 3}(F)$. Show that ...
1
vote
2answers
68 views

Let $A$ be a complex $2$ by $2$ matrix having distinct eigenvalues $a, b$. Show that $A^n =\frac{ a^n}{a - b}(A - bI) + \frac{b^n}{b - a}(A - aI)$.

Let $A\in\mathscr{M}_{2\times 2}(\mathbb{C})$ be a matrix having distinct eigenvalues $a\neq b$. Show that, for all $n > 0$, \begin{equation*} A^n =\frac{ a^n}{a - b}(A - bI) + \frac{b^n}{b - ...
3
votes
1answer
87 views

Characterize magic matrices in terms of their eigenvalues. A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$.

A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$. Characterize magic matrices in terms of their eigenvalues. (Exercise 705 from Golan, The Linear Algebra a ...
1
vote
1answer
48 views

Let $D$ be a nonsingular diagonal matrix. Show that $1\notin spec(DA)$ if and only if $D - A$ is nonsingular.

Let $F = \mathbb{F}_3$ and let $n$ be a positive integer. Let $D = [d_{ij} ]\in\mathscr{M}_{n×n}(F)$ be a nonsingular diagonal matrix and let $A\in\mathscr{M}_{n×n}(F)$. Show that ...
1
vote
0answers
42 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
2answers
67 views

Do positive-definite matrices always have real eigen values?

Do positive-definite matrices always have real eigenvalues? I tried looking for examples of matrices without real eigenvalues (they would have even dimensions). But the examples I tend to see all ...
3
votes
4answers
102 views

Finding a matrix representation of the transpose transformation

Define $T : M_{n×n}(\mathbb{R}) → M_{n×n}(\mathbb{R})$ by $T(A) := A^t$. I know this transformation is linear and just takes a matrix and spits out it's transpose. I also know that the transpose is ...
13
votes
4answers
248 views

Matrices such that $M^2+M^T=I_n$ are invertible

Let $M$ be an $n\times n$ real matrix such that $M^2+M^T=I_n$. Prove that $M$ is invertible Here is my progress: Playing with determinant: one has $\det(M^2)=\det(I_n-M^T)$ hence ...
4
votes
2answers
75 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, ...
3
votes
1answer
116 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 $tr[(AB)^2]\le tr(A^2B^2)$?
3
votes
3answers
1k views

$\operatorname{adj}(AB) = \operatorname{adj} B \operatorname{adj} A$

How can I prove that $\operatorname{adj}(AB) = \operatorname{adj} B \operatorname{adj} A$, if $A$ and $B$ are any two $n\times n$-matrices. Here, $\operatorname{adj} A$ means the adjugate of the ...
1
vote
1answer
67 views

Checking whether the vectors are linearly independent.

I just finished an assignment and I would like someone that knows this material to basically check if I'm doing it right. I was given 3 vectors <1,2,3>, <1,0,1>, <2,1,0> in R^3 and need to ...
0
votes
4answers
59 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.
3
votes
3answers
97 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
47 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
93 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 ...
2
votes
1answer
82 views

Absolute value of eigenvalues of a $3 \times 3$ matrix

Let $$A = \left[ {\begin{array}{cc} 1 & 1 & 1 \\ 1 & w^2& w \\ 1 & w & w^2 \end{array} } \right] $$ where $w$ is a cube root of unity (other than 1). Let ...
1
vote
0answers
66 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 ...
0
votes
0answers
20 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 ...
1
vote
0answers
34 views

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

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

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$?

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$? I read a book a very long time ago where it said something like this: Given a system of two equations with two ...
1
vote
0answers
19 views

An equality between maximums of two logdet expressions

I have the following question. Let $K$ be a positive-definite $N\times N$ real-valued matrix (I'll denote this by $0\prec K$ and will subsequently assume all matrices are $N\times N$ and real-valued) ...
1
vote
0answers
51 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
3answers
110 views

Determinant properties

Prove without expanding: \begin{equation}\begin{vmatrix}1&1&1\\a^2&b^2&c^2\\a^3&b^3 & c^3\end{vmatrix} = (ab + ac + bc)(b - a)(c - a)(c - b)\end{equation} I tried to zero some ...