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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4
votes
2answers
159 views

How prove this $det\left(\frac{1}{\lambda^2_{i}+t\lambda_{i}\lambda_{j}+\lambda^2_{j}}\right)_{n\times n}>0,-2<t<2$

Question: Show that for $t\in (-2,2)$ and $0<\lambda_1<\lambda_2<\ldots<\lambda_n$ we have $$det(A)=det\left(\dfrac{1}{\lambda^2_{i}+t\lambda_{i}\lambda_{j}+\lambda^2_{j}}\right)_{n\times ...
2
votes
1answer
311 views

rank one update for Cholesky factor

I have covariance matrix known to be $$K = \sum_{i=1}^Nx_ix_i^T$$ where the dimension of $x$ is big (like 50000) so I don't want to really compute any outer-product to expand it as a full matrix. ...
3
votes
1answer
119 views

An invertible matrix

Given Matrix $A$, checking that its diagonal elements are nonzero or whether its determinant is nonzero, can we say the matrix is invertible for sure? Are there other properties that by looking at the ...
2
votes
1answer
51 views

How find this $M(2)$

let matrix $$A(x)=\begin{bmatrix} 1&x\\ x&1 \end{bmatrix}$$ and consider the infinte matrix product $$M(t)=\prod_{n=1}^{\infty}A(p^{-t}_{n})$$ where $p_{n}$ is the nth prime Evaluate $M(2)$ ...
1
vote
2answers
36 views

Finding associated eigenvalue and eigenvector

Having troubles with this question Suppose that $\det(A) \not= 0$, and $A$ and $B$ both have eigenvector $v$, but the corresponding eigenvalue is $\lambda_{A}$ for $A$ and $\lambda_{B}$ for $B$. Show ...
-1
votes
2answers
394 views

Proof that $\det(A) = 0$ implies linear dependence of columns of a matrix $A$ [closed]

Let $A$ be an $n \times n$ matrix. How would you rigorously prove that $\det(A) = 0$ if and only if the columns of $A$ are linearly dependent?
4
votes
2answers
410 views

Characterizing sums of permutation matrices

Given an $n$ by $n$ matrix $A$ whose rows and columns sum to $m \in \mathbb N$ and entries are nonnegative integers, does there exist a permutation matrix $P$ such that $A - P$ has only nonnegative ...
1
vote
1answer
50 views

Does the identity matrix adapt to any other matrix?

So I have a matrix of the form $X=AX+B$ Where $X$ is a 3 by 1 column matrix, $A$ is a 3 by 3 matrix and $B$ is a 3 by 1 column matrix. (Notice that I am talking about Leontief input-output). So I ...
9
votes
3answers
975 views

Relationship between Nilpotent Matrix and Matrix with all zero diagonal factors.

solving Linear Algebra HW, I suddenly became curious about the relationship between Nilpotent Matrix and matrix with all zero diagonal factors such that $A_{11} = A_{22} = \cdots = A_{nn} = 0$ Does ...
3
votes
1answer
51 views

When solving via gauss-jordan

When I solve via Gauss-Jordan, taking a $3 \times3$ matrix as an example... Should I always try and get a $1$ in the upper left corner and $0$'s in the rest of the column followed by getting a $1$ in ...
1
vote
2answers
122 views

characterization of uniform ellipticity

Let $B$ be a $n\times n$ matrix over $\mathbb{R}$ and define $A:=BB^*$. I read in a paper that the following two statements are equivalent: (1) the matrix $A$ is uniformly elliptic; i.e. for all ...
3
votes
1answer
40 views

name of matrix of inner products $\langle f_i, f_j\rangle$

Given a Hilbert space $H$ and a number of elements $\phi_i\in H$, does the matrix $M$ with $$ M_{i,j} := \langle\phi_i, \phi_j\rangle $$ have any particular name?
1
vote
1answer
285 views

Prove scalar products are invariant under all orthogonal transormation

I wondering how to prove: That scalar products are invariant under all orthogonal transformation: $<\!x, y\!>\; =\;<\!Qx, Qy\!>$ which holds for all vector $x$,$y \in \Re^n$ and all ...
3
votes
3answers
76 views

Commutation of exponentials of matrices

Given two $n \times n$ real matrices $A$ and $B$, prove that the following are equivalent: (i) $\left[A,B\right]=0$ (ii) $\left[A,{\rm e}^{tB}\right] = 0,\quad$ $\forall\ t\ \in\ \mathbb{R}$ (iii) ...
1
vote
0answers
69 views

Why the pivots and relative to the eigenvalue in symmetric matrix?

In the book, it said, there a quick fast way to test whether the eigenvalue are all positive or not. Just check the pivot of the symmetric matrix, if x no. of positive pivot, it would have x no.of ...
-2
votes
1answer
66 views

Determine $b$ where the system has a solution

Determine the value of $b$ for which the system $$\begin{align} x_1 + 4x_2 − 3x_3 + 2x_4 &= 2\\ 2x_1 + 7x_2 − 4x_3 + 4x_4 &= 3\\ −x_1 − 5x_2 + 5x_3 − 2x_4 &= b\\ 3x_1 +10x_2 − 5x_3 + ...
5
votes
2answers
264 views

Finding the determinant of a matrix

I am given $A = \begin{pmatrix} a & b\\ c & d \end{pmatrix} $ and B = $ \begin{pmatrix} e & f\\ g & h \end{pmatrix}$ whose elements are non-zero reals. If $BA = I$, where ...
2
votes
1answer
63 views

the rank of matrix products including a commutation matrix

Given a full rank matrix $A \in \mathbb{R}^{M \times N^2}$ where the rank of ${A}$ is ${\rm rk}(A)= M \leq N^2$ and the commutation matrix $K_{NN}$. I need to find the rank of a matrix product ...
1
vote
2answers
73 views

Eigenvalues of a rank-one update to a rank-one matrix

Let $\mathbf{a}$ and $\mathbf{b}$ be two column vectors in $\mathbb{C}^N$. What can we say about the eigenvalues of the matrix \begin{align} \mathbf{a}\mathbf{a}^H+\mathbf{b}\mathbf{b}^H \end{align} ...
0
votes
1answer
142 views

Is there a simple method to finding orthonormal basis given a partially complete set

I have a question Find the indicated projection matrix for the given subspace, and find the projection of the indicated vector $<2,-1,3>$ on ...
0
votes
1answer
52 views

Can you find an expression for $F_{12}(t,\tau)$ in terms of $F_{11}(t,\tau)$ and $F_{22}(t,\tau)$?

I have a problem with this...I can not figure out how to solve it..! can you help me? thank you!! Show that if $A(t)$ is partitioned as $$ A(t) = \begin{pmatrix} A_{11}(t) & A_{12}(t) \\ ...
3
votes
2answers
108 views

How find this matrix value of this $\det(A_{ij})$

Find this value $$\det(A_{n\times n})=\begin{vmatrix} 0&a_{1}+a_{2}&a_{1}+a_{3}&\cdots&a_{1}+a_{n}\\ a_{2}+a_{1}&0&a_{2}+a_{3}&\cdots&a_{2}+a_{n}\\ ...
1
vote
4answers
74 views

Finding $A^k$ for non-diagonalizable $A$

Is there an easy way to find $A^k$ for a square matrix $A$ that is NOT diagonalizable?
0
votes
2answers
123 views

Calculating the norm of a matrix in matlab

So I've been given the matrix A A = [1 2; 3 1] and the question is: If v is an error vector with ||v|| = 0.01, give an upper bound on ||A^30v||. ||A^30v|| = ||A^30|| * ||v||, correct? I used the ...
1
vote
2answers
79 views

Show that the matrix is invertible

let $A \in M_n(F)$ be a n by n matrix with values from an unknown field $F$. $P_A(t)$ is the characteristic polynomial of $A$, and $g(t) \in F[t]$ a polynomial of an unknown degree. assume that ...
0
votes
1answer
167 views

Inverse of a certain block matrix

So I'm trying to compute the inverse of a block matrix that's a subset of a larger consideration I was attempting (this particular matrix comes from the normal and orthogonal equations for least ...
1
vote
2answers
73 views

Unitary map between sets of vectors

Suppose I have two sets of vectors, $E_1=\{v_i\}_{i=1}^{k}$ and $E_2=\{u_i\}_{i=1}^{k}$, with each vector belonging to $\mathbb{C}^k$. When is it possible to find a unitary matrix that maps $E_1$ to ...
0
votes
1answer
34 views

Show that a transformation matrix is equal to the martix of eigenvectors

The real symmetric $3\times 3$ matrix $A$ has unit eigenvectors $\mathbf x_i$, $i=1,2,3$. Thus we have, $A\mathbf x_i=\lambda_i \mathbf x_i$. A $3\times 3$ matrix $C$ takes a vector in the ...
1
vote
3answers
198 views

Square root of nilpotent matrix

How could I show that $\forall n \ge 2$ if $A^n=0$ and $A^{n-1} \ne 0$ then $A$ has no square root? That is there is no $B$ such that $B^2=A$. Both matrices are $n \times n$. Thank you.
1
vote
1answer
67 views

Question regarding $A = B^{-1}DB$ and determinants

Consider $A = B^{-1}DB$, where $A$ is a normal matrix represented by unitary matrices $B, B^{-1}$ and the diagonal matrix $D$. Although $B^{-1}B = BB^{-1} = I_B$ why doesn't $B^{-1}DB$ give you $D$? ...
1
vote
2answers
32 views

Geometric Interpretation of members of $\mathrm{O}(2)\setminus\mathrm{SO}(2)$

I recently came across a question which asked to prove the defining properties of the orthogonal matrices (members of $\mathrm{O}(2)$), then to subsequently determine that they can be written in the ...
0
votes
3answers
55 views

Has this system unique solutions?

Can i find solutions for this system of equations? $x\cdot y= a$ $\frac{x}{y}= b$ in which x,y are unknown and a,b known values? My only hesitation is that the system has many solutions and the ...
5
votes
2answers
230 views

$R$ has only one maximal ideal

Let $F$ be a field. Let $R$ be the set of all upper triangular matrices of the ring $M_{n}(F)$ with the property that its coefficients on the main diagonal are all the same. Prove that $R$ has only ...
3
votes
2answers
99 views

Proof that $(\alpha I - A)$ invertible if $\alpha > \rho(A)$

I want to proof that for $A \in \mathbb{R}^{n \times n}$ with $a_{ij}\geq 0, \forall i,j=1,...,n$: \begin{align} (\alpha I - A) \text{ is invertible if } \alpha > \rho(A) \end{align} where ...
1
vote
1answer
27 views

Finding a matrix from an equation given

If $M$ is given in $L'*L^2*L'=M$, can we find $L$? If so how? We know that $M$ is positive definite matrix. I am working on image processing and solving this challenge can bring several ideas.
1
vote
1answer
66 views

Resolvent recurrence relation

Let the resolvent matrix of $\mathbf{X}$, a symmetric matrix with real entries, be defined as \begin{align} R_{\mathbf{X}}(\lambda):=\bigl(\mathbf{X}-\lambda\mathbf{I}\bigr)^{-1}, \qquad \lambda ...
3
votes
1answer
139 views

Does 1 distinct eigenvalue guarantee 1 eigenvector?

I am trying to figure out when 2x2 matrices are not diagonalizable. Right now, my conditions are: the matrix has only 1 distinct eigenvalue the matrix yields only 1 linearly independent eigenvector ...
0
votes
1answer
90 views

Let $L : \mathbb{R}^2 \to \mathbb{R}^2 $ be a linear transformation whose matrix in the standard basis is

Let $L : \mathbb{R}^2 \to \mathbb{R}^2 $ be a linear transformation whose matrix in the standard basis is $$\mathrm{A}= \begin{bmatrix} 2 & -1 \\ 3 & 6 \\ ...
2
votes
4answers
2k views

$3\times 3$ matrix with no real eigenvalues

I was asked this question on my hw along with any $2\times2$ matrix with no real eigenvalue and any $4\times4$ matrix with no real eigenvalue. I got the $2\times2$ which is $$ \begin{bmatrix} 1 & ...
1
vote
2answers
274 views

How to diagonalize a matrix

So I am trying to diagonalize this matrix {2,0,-2} {1,3,2} {0,0,3} so that those are the rows of the matrix. I know the eigen values are 2 and 3. I don't think that this matrix can be ...
0
votes
1answer
32 views

Matrix question regarding symmetric

I need help with the following problem Express the matrix $$B=\begin{pmatrix} 2 &-2 &-4 \\ -1 &3 &4\\ 1 &-2 &-3 \end{pmatrix}$$ as the and sum of a symmetric and a skew ...
1
vote
2answers
83 views

Is matrix multiplication really a group operator?

A group has an operation that can be performed over ANY two elements in a set. Given that an $n \times m$ matrix can only be multiplied by an $m \times o$ matrix, doesn't that mean that matrix ...
2
votes
3answers
531 views

What is the difference between Symmetric vs Skew Symmetric?

I want to know the difference between Symmetric Symmetric vs Skew Symmetric?
1
vote
0answers
76 views

How to find derivative of matrix?

I'm reading a statistical paper by Breslow and Clayton (1993) and there is a section: ... , we use in practice the REML version (Patterson and Thompson 1971): ...
1
vote
1answer
43 views

Why, while checking consistency in $3\times3$ matrix with unknowns, I check only last row?

I would like to know whether my thinking is right. So, having 3 linear equations, $$ \begin{align} x_1 + x_2 + 2x_3 & = b_1 \\ x_1 + x_3 & = b_2 \\ 2x_1 + x_2 + 3x_3 & = b_3 \end{align} $$ ...
5
votes
3answers
3k views

Proving that a right (or left) inverse of a square matrix is unique using only basic matrix operations

Proving that a right (or left) inverse of a square matrix is unique using only basic matrix operations -- i.e. without any reference to higher-order matters like rank, vector spaces or whatever ( ...
0
votes
1answer
80 views

Find the Characteristic polynomial

The characteristic polynomial of $A \in M_{4}(\Bbb R)$ is: $P(t)= t^4-t$ Find the Characteristic polynomial of: $A^2, A^4$ ($A^4$ was easy but with $A^2$ I'm stuck) Same question with the field ...
0
votes
1answer
74 views

Invertible matrices

Please help with the next question. Let A and B be two invertible matrices such that $A+B \neq 0$. Prove or disprove that A+B is invertible. Thanks!
7
votes
2answers
353 views

Matrices which commute with all the matrices commuting with a given matrix [duplicate]

Let $A$ be an $n \times n$ matrix with entries from an arbitrary field $F$ and let $C(A)$ denote the set of all matrices which commute with $A$. Is it true that $C(C(A))= \{ \alpha_1 + \alpha_2 A ...
0
votes
1answer
24 views

need help in figuring out equation & graph for a matrix

Let $A$ be a $2\times2$ matrix defined by $$A=\begin{bmatrix}3 & 0\\0 & 2 \end{bmatrix},$$ and $(x,y)$ satisfy an equation $x^2+y^2=1$. If $(x',y')$ is the image of $(x,y)$ under the matrix ...