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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0
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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 ...
2
votes
4answers
63 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
0answers
3 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
votes
0answers
8 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
9 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
7 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 ...
1
vote
2answers
22 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
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1answer
73 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 ...
-2
votes
1answer
28 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
37 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
vote
0answers
30 views

Matrix pencils of quadratic forms

Consider a matrix pencil of quadratic form $F-λB$ with $B$ positive definite. For which $λ$ the pencil $F-λB$ 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 $T : V → V$ 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 $S : V → V$ ...
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
40 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
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0answers
10 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. (s−1)x +(s+3)y + z = 1 s x ...
0
votes
2answers
45 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
votes
0answers
23 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
49 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? ...
0
votes
0answers
15 views

cofactor expansion

I want to find the determinant of the following matrix using cofactor expansion: ${ \begin{matrix} 1 & 1 & 1 \\ 2 & 3 & 7 \\ 6 & 8 & 9 \\ \end{matrix} }$ So I am going to use ...
2
votes
0answers
22 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
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0answers
28 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
37 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
8 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
17 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
18 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) ϵ R if and only if x-y=0. If I am ...
-5
votes
1answer
43 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
26 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
55 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
22 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
vote
0answers
8 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
50 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
62 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
39 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 ...
5
votes
1answer
27 views

Exist basis, simultaneously upper-triangular?

Let $A, B \in M_n(\mathbb{C})$ be such that $\text{rank}(AB - BA) \le 1$. Does there exist a basis of $\mathbb{C}^n$ with respect to which $A$ and $B$ are simultaneously upper-triangular?
0
votes
2answers
35 views

$f$ is a differentiable map and compute $Df(A)(H)$.

Let $f : GL(n, \Bbb R) \to GL(n, \Bbb R)$ be defined by $f(A) = A^{-1}$ where derivative of the matrix $A$ exists. Then $f$ is a differentiable map and compute $Df(A)(H)$. $A A^{-1} = I \implies ...
0
votes
1answer
27 views

If $A \in M_n(R)$, with $R$ a P.I.D., can $A$ be put in Jordan form iff all the roots of the characteristic polynomial are in $R$?

If $A \in M_n(R)$, with $R$ a P.I.D., can $A$ be put in Jordan form iff all the roots of the characteristic polynomial are in $R$? If this is false in general, is it possibly true for nilpotent ...
1
vote
1answer
44 views

About a matrix identity.

In a document named as "The Matrix Cook-Book" I saw two expressions of which I do not get any clue how they are derived. For $n = 3:$ $\det(I + A) = 1 + \det(A) + Tr(A) + 1/2\ Tr(A)^2 − 1/2\ ...
0
votes
0answers
32 views

Matrices and determinant.. [on hold]

Use elementary row operations to evaluate |A|, and then evaluate A = $$ \left[ \begin{array}{cc|c} 1&2\\ 4&5 \end{array} \right] $$ Find |(AA^T)^2| ? can anyone tell me the ...
5
votes
1answer
31 views

Finding an explicit eigenvector

Let $A$ be an $n\times n$ matrix over a field and let $\operatorname{adj}(A)$ denote its classical adjoint. Suppose all column sums of $A$ are zero so that $A$ is singular. If $\operatorname{rank}(A) ...
0
votes
0answers
13 views

Geometrical interpretation of the condition number as measure of matrix dissimilarity

Consider two $p$ by $p$ symmetric positive definite matrices $\pmb F$ and $\pmb G$ and denote $$\pmb D=\pmb G^{-1/2}\pmb F \pmb G^{-1/2}.$$ Sometimes, the condition number of $\pmb D$ will be used ...
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vote
2answers
51 views

Distinct eigenvalues and matrices problem

Let $V$ be a real vector space and $T: V \rightarrow V$ be a linear transformation. It is given that if $v_1, . . . , v_n$ are eigenvectors for distinct eigenvalues $λ_1, . . . λ_n$ then $\{v_1, . . ...
2
votes
1answer
19 views

Equality of determinants for a specific collection of square matrices of size $n=2^m$

My investigations have led me to a question that I am convinced is true. I need to show that, for a given $m$, a certain collection of square $n=2^m$ matrices have the same determinant. In dimension ...
0
votes
0answers
15 views

Tensors, indices and matrix notation - is there a common convention?

For a tensor named T with two indices, there are four possibilities: $T_{ij}$ , $T_i^{\ j}$, $T^i{\ _j}$ and $T^{ij}$. Is there a common convention as to how these tensors would be represented as ...