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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0answers
22 views

A question about product of three positive definite matrices

Assume that $A,B$ and $C$ are symmetric positive definite matrices. I guess that the eigenvalues of the matrix $D=ABC$ can be any complex numbers. Is that true?
0
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1answer
14 views

Matrix polynomials/eigenvalues

$\begin{pmatrix} 7 & -2\\2 & 2 \end{pmatrix}$ The eigenvalues for this matrix are $\lambda=6$ and $\lambda=3$ It also happens that $(A-6I)(A-3I)=0$ I've checked for various $2$ x $2$ ...
-1
votes
0answers
12 views

A question on matrix norm

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
-1
votes
1answer
37 views

Three lines that intersect in a plane.

Find a condition for three lines (š‘– = 1,2,3) in a plane given by $š‘Ž_š‘– š‘„ + š‘_š‘– š‘¦ = š‘_š‘–$ to intersect in one point. I decided to form a matrix and to find the identity matrix since it will ...
1
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0answers
11 views

Property of hermitian matrices (eigen values)

In a paper I have read the following ${\bf G}$ is a Hermitian matrix, that 1) ${\bf G}$ is diagonalizable 2) the singluar values are same as the eigen values Is number 2 correct? I cant seem to ...
0
votes
1answer
21 views

Is there any inner product on $M_{n \times n}$ inducing this norm?

The set $M_{n \times n}$ is the collection of all $n \times n$ matrices over $\mathbb{R}$. Definition: $\|A\|_2=Sup_{\|u\|_2=1} \|Au\|_2$. Is there any inner product on $M_{n \times n}$ inducing ...
2
votes
1answer
32 views

A question in matrix norm.

Suppose $A \in {\mathbb C^{n \times n}}$ and $\left\| A \right\| \le \varepsilon $ $v \in {\mathbb C^n}$ and ${v^*}v = 1$ Is this true that $\left\| {{v^*}Av} \right\| \le \varepsilon $?
0
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0answers
8 views

Response Matrix with finit actuator

I have a system of penalties ($P$) and actuators($A$). Whereby: d$P_i/$d$A_j$ = close to constant $\quad\forall i,j$ In order to minimize $P$, I create a response Matrix ($M$). With its ...
0
votes
1answer
24 views

A matrix with one non-zero singular value

I have a question regarding matrices and eigen values. If SVD decomposition was performed on matrix, and the inner matrix of singular values has only one non zero value. Should the left and right ...
0
votes
1answer
27 views

A question on spectrum [duplicate]

Let $A,B \in {C^{n \times n}}$ and ${\sigma (A + B)}$ is spectrom of $(A+B)$. Suppose $M = \left\{ {\lambda \in C:\lambda \in \sigma (A + B),\left\| B \right\| \le \varepsilon } \right\}$ $F(A) = ...
0
votes
1answer
17 views

In a matrix does every set of r row vectors need to be linearly independent for rank to be r?

Rank of a matrix is the maximum number of linearly independent row vectors , does every set of r row vectors need to be linearly independent or finding only one set of r row vectors which are linearly ...
3
votes
4answers
788 views

Why is the matrix product of 2 orthogonal matrices also an orthogonal matrix?

I've seen the statement "The matrix product of two orthogonal matrices is another orthogonal matrix. " on wolframs website but haven't seen any proof online as to why this is true. By orthogonal ...
0
votes
1answer
19 views

Show that if the leading principal minors of a nonsingular $n\times n$ matrix $A$ are all nonzero then the matrix $A$ has $LU$ factorization

I am stucked at this problem: Prove by induction that if the leading principal minors of an $n\times n$ nonsingular matrix $A$ are all nonzero then the matrix $A$ has $LU$ factorization. (The ...
0
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0answers
19 views

Show that for every $n\times n$ matrix $A$, there exist an $n\times n$ permutation matrix $P$ such that $PA$ has $LU$ factorization

I am stucked at this problem: Prove by induction on $n$ that for every $n\times n$ matrix $A$, there exist an $n\times n$ permutation matrix $P$ (a matrix obtained by rearranging the rows (or ...
0
votes
0answers
14 views

finding all $m\times k$ matrices with prescribed row and column sums and zero elements

I'm looking for an algorithm constructing non-negative integer matrices with prescribed row and column sums and some predefined zero entries. For example, if column sums are [1 1 2 1 1] and row sums ...
0
votes
0answers
10 views

Question regarding Eigen Value Decomposition and Singular Value Decomposition

I have a product of matrices that have the following form $$ {\bf A} ^H {\bf A}$$ where subscript $H$ means hermitian transpose. I am trying to find the eigen value decomposition (EVD) of ${\bf ...
0
votes
0answers
17 views

Parametric vector form of cartesian equation

Cartestian equation: $$-2x-y+z=6$$ I know to find the parametric vector form we can find any 3 points P, Q and R which satisfy the cartesian equation. $$ \begin{pmatrix} x_1\\ y_1\\ z_1 ...
1
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2answers
29 views

How to solve for the matrix $X$ in the following equation $AXB + X = CD$

How to solve for the matrix $X$ in the following equation $AXB + X = CD$? $A$ and $B$ are full rank symmetric matrices, and there is no structure to $CD$. $CD$ just could be $C$.
0
votes
1answer
76 views

Is it true a matrix $A$ has determinant $0$ if and only if $A^N=0$?

I know that the determinant doesn't stay the same for a matrix $A$ for which the determinant $\neq 0$. I just calculated some determinants of a $3\times 3$ matrix to find that out. But I also ...
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votes
1answer
44 views

A question on numerical range

Let $A,B \in {C^{n \times n}}$ and ${\sigma (A + B)}$ is spectrum of $(A+B)$. Suppose $M = \left\{ {\lambda \in C:\lambda \in \sigma (A + B),\left\| B \right\| \le \varepsilon } \right\}$ $F(A) = ...
4
votes
0answers
42 views

Upper bound on infinity norm of inverse of a positive definite matrix

Consider a positive definite matrix, $A$, and the following quantity: \begin{align} \|A^{-1}\|_\infty \end{align} Are there any upper bounds on the above normed term?
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0answers
23 views

How do you expand a matrix to a power?

Suppose I have an nxn matrix A, where t is a natural number >0. Is A^t=A^(t-1)A or A^t=AA^(t-1) I would think that the operation of splitting them up into these two should work. However, A and ...
0
votes
1answer
30 views

Does this matrix operation hold?

Suppose A is an nxn matrix and b is a constant scalar. t is some natural number >0 Can i apply binomial expansion on (A-Ib)^t?
1
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0answers
33 views

Matrix pencils of quadratic forms

Consider a matrix pencil of quadratic form $F-Ī»B$ with $B$ positive definite. For which $Ī»$ the pencil $F-Ī»B$ less or equal to $0$ (negative definite)?
0
votes
0answers
15 views

Matrices of Ordered Bases

Let $V$ be a real finite-dimensional vector space and $T : V ā†’ V$ be a linear map. Let $E$ be a basis of V . What does it mean to say that $A$ is the matrix of $T$ with respect to $E$. Let $S : V ā†’ V$ ...
2
votes
1answer
60 views

Sign of $tr(A)$ given $I_n+A+A^2+A^3=0$

Let $A$ be a real matrix such that $I_n+A+A^2+A^3=0$, what is the sign of $tr(A)$ ($tr$ being the trace) ? What I have done : One can easily figure our the inverse of $A$ since ...
1
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2answers
48 views

Relation between norms of two matrices

Is there a relation between the norm $\|A\|$ of a nonsingular symmetric positive definite matrix $A$ and the norm of its inverse matrix $A^{-1}$?
0
votes
0answers
12 views

Augmented Matrix and Row echelon form

For which real numbers s and t does the following linear system have (a) no solution, (b) exactly one solution, or (c) infinitely many solutions? Justify your answers. (sāˆ’1)x +(s+3)y + z = 1 s x ...
0
votes
2answers
49 views

generalized Cauchy-Schwarz inequality

How to prove $A'B(B'B)^{-1}B'A \leq A'A$, where $A$,$B$ are $n\times k$ matrices and $B'B$ is assumed to be positive definite? I don't see why it is a Cauchy-Schwarz inequality.
0
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0answers
25 views

Is there any smart way to check triangle inequality for a matrix?

Here is the description of the problem: We have a matrix with: all (i,i) cells are 0; Some cells are filled with certain number while others are left blank. Now, we want to fill the blanks with ...
3
votes
2answers
52 views

Can such an “orthogonal” matrix exist?

I know that the definition of an orthogonal matrix is that $A \in \mathbb R^{n \times n}$ is orthogonal if $AA^T = A^T A=I$, no problem with that whatsoever. My question is this - Why only square ...
0
votes
1answer
21 views

vector matrix division

I can multiply a vector by a matrix like so a d e f ad + be + cf b * g h i = ag + bh + ci c j k l aj + bk + al but how do I divide? ...
2
votes
0answers
23 views

Problem with determinant

Let $A\in\mathbb{C}^{3\times 3}$ and $x,y\in\mathbb{C}^3$. Prove that $det\left(I-\frac{xy^*A}{1+y^*Ax}\right)=\frac{1}{1+y^*Ax}$ How can I prove this?
3
votes
0answers
34 views

Matrix product bound

Consider the following inequality \begin{align*} AB^{-1}A^\top \preceq cI \end{align*} where $A\in\mathbb{R}^{n\times m}$, $B\in\mathbb{R}^{m\times m}$, $c\in\mathbb{R}$ (given), and $I$ is the ...
1
vote
1answer
38 views

An equivalent definition of the condition number of a matrix [on hold]

How can I prove that the condition number can't be expressed by $$\kappa(A)= \sup_{\lvert\lvert x \rvert \rvert=\lvert \lvert y \rvert \rvert} \lvert\lvert Ax\rvert \rvert/\lvert\lvert Ay\rvert ...
2
votes
0answers
9 views

Matrices with left and right singular vectors being vandermonde matrices

Assume we have matrices ${\bf H_i}$ for $i\in[1:K]$ and that the Singluar Value Decomposition (SVD) of ${\bf H_i}$ is such that $${\bf H_i = A_{bi} D_iA_{si}^*}$$ where ${\bf A_{bi}}$ and $ {\bf ...
0
votes
1answer
21 views

Is the spectral radius of a matrix a convex norm of it?

I am wondering if the spectral radius of a matrix is may be some kind of a norm ($l_{\infty}$-norm?) of it and if that is convex. Any pointers to related ideas would be helpful too.
0
votes
1answer
19 views

Matrix and eigenvalues question hints?

This is the homework I have done part a, b, but I don t have any idea how to do the rest $y = 5$ and $z = 12 $ Those are the eigenvalues of matrix $A$ For part c, and d, I've tried to put some ...
0
votes
1answer
10 views

Is this relation considered antisymmetric and transitive?

I'm having trouble understanding whether or not this relation would be considered antisymmetric and transitive. The a relation R on the set of real numbers by (x,y) Ļµ R if and only if x-y=0. If I am ...
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votes
1answer
44 views

Using matrices to solve questions [on hold]

A certain library owns 10 000 books. Each month 20% of the books in the library are lent out and 80% of the books lent out are returned, while 10% remain lent out and 10% are reported lost. Finally, ...
0
votes
1answer
28 views

How does permutation works in “multimatrices”?

I want to adequately define a $m\times n$ "multimatrix" that satisfies these properties: 0.A $m\times n$ multimatrix has $m\times n$ entries just like a normal matrix. It is the positions they occupy ...
2
votes
6answers
56 views

For $n\times n$ matrices, is it true that $AB=CD\implies AEB=CED$?

If $A,B,C,D,E$ are $n\times n$ matrices, does $AB=CD$ imply $AEB=CED$? I only know that $AB=CD \implies ABE=CDE$, but I don't see how you can sandwhich $E$ within it. Also, if $AB=CD=0$, does ...
1
vote
2answers
60 views

Matrix with all 1's diagonalizable or not? [on hold]

This is a followup to my question here. Let $A$ be the $n \times n$ matrix over a field of characteristic 0, all of whose entries are 1. Is $A$ diagonalizable?
0
votes
1answer
33 views

Diophantine equation by matrice?

I want to learn how solve simple ax+by=c with matrices (assuming that's the fasted method?), but it's difficult to find correct learning material. I've been through this process: 4386x + 89744y ...
0
votes
0answers
7 views

Eigen-decomposition of augmented block rectangular matrix

I have a rectangular matrix $\mathbf{X}_{n\times p}$ where the eigenvector decomposition of its inner product with itself is $$ \mathbf{X}^T\mathbf{X} = \mathbf{P}^T\mathbf{\Lambda P} $$ where ...
0
votes
2answers
23 views

Matrix exponential question

Wiki https://en.wikipedia.org/wiki/Matrix_exponential said: if a matrix A is diagonal $$A=\begin{bmatrix} a_1 & 0 & \ldots & 0 \\ 0 & a_2 & \ldots & 0 \\ \vdots & \vdots ...
1
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0answers
9 views

Question on applications using schur complements

i wonder if you may be able to contribute some areas/ideas where the use of schur complements are used. Like for exampple, I think schur complements can be used to check for positive definiteness of ...
2
votes
1answer
51 views

$A^k = I$ implies diagonalizable? [duplicate]

If $A$ is a square complex matrix with $A^k = I$ (where $I$ is the identity matrix of the same size as $A$) for some positive integer $k$, does it follow that $A$ is diagonalizable?
5
votes
3answers
65 views

$\mathbb{C}$-algebra automorphism of $M_n(\mathbb{C})$ has form $X \mapsto AXA^{-1}$.

As the title suggests, what is the easiest way to see that any $\mathbb{C}$-algebra automorphism of $M_n(\mathbb{C})$ has the form $X \mapsto AXA^{-1}$ for some fixed $A \in GL_n(\mathbb{C})$?
0
votes
2answers
40 views

Finding eigenvalues and eigenvectors of $2 \times 2$ matix

I having a few issues finding the eigenvectors for the following matrix: $$ \begin{bmatrix} -1 & -1\\ 0 & -2 \\ \end{bmatrix}$$ I calculated the eigenvalues to be ...