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), ...

learn more… | top users | synonyms (2)

0
votes
1answer
43 views

vectors and matrix problem

Let $A$ be a $5$x$5$ matrix, with the third column of $A$ represented by $a_3$, and let $b$ be a $5$x$1$ non-zero column vector. Suppose that the matrix equation $Ax=b$ has a unique solution x, with ...
0
votes
0answers
24 views

Matrix problem, subspace

Suppose you are given a matrix A and have calculated an echelon form R of A. (Note: R is not assumed to be in reduced row echelon form.) Which of the following statements must be true? (Select all ...
-1
votes
1answer
21 views

A basic question about eigenvalue

Suppose a symmetric matrix $A$ is of dimension $N \times N$. Then the largest eigenvalue of $A$ is equal to $\max_{i} \sum^{N}_{j=1} |A_{ij}|$. Is this statement true? If so, how shall I show it ...
0
votes
2answers
22 views

Is the following set of vectors in $\Bbb R^3$ linearly dependent?

I am using Anton's Elementary Linear Algebra book (8e) and trying to do exercise set 5.3, question 2a It gives the vectors $(4,-1,2)$, $(-4,10,2)$ and asks if they are linearly dependent . My final ...
3
votes
1answer
18 views

Determinant of block matrix with commuting blocks

I know that given a $2N\times 2N$ block matrix with $N\times N$ blocks like $\mathbf{S} = \begin{pmatrix} A & B\\ C & D \end{pmatrix}$ we can calculate ...
0
votes
1answer
38 views

Showing a $2\times2$ matrix is invertible

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there exists a two-dimensional vector ${w}$ such that ${A} {w} = {v}.$ Show that ${A}$ is invertible. I have no idea on how ...
2
votes
1answer
24 views

Checking psd-ness of matrix

I have the following problem and don't know how to proceed... I want to check if \begin{equation} \frac{1}{2}(B^\top A^\top A + A^\top A B) - \frac{1}{4}B^\top A^\top A A^\top (AA^\top ...
3
votes
1answer
33 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. I know that $c$ is an egenvalue and $[1,...,1]^T$ is ...
1
vote
1answer
12 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 ...
0
votes
2answers
16 views

Reflect on y axis in 3D Matrix?

I have a question saying "Define a 3D Matrix that performs a reflection in the y axis" but I don't know how to solve it. So if we have a 2D matrix and we say 'reflection on the y axis' we mean that x ...
0
votes
3answers
12 views

Give two matrices whose column spaces contain the column space of the given matrix.

Let $$B = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\text{.}$$ Give two matrices whose column spaces contain $C(B)$, the column ...
2
votes
2answers
24 views

Orthogonal projection and subspaces

Consider the vector space $\mathbb{R}^m$ with usual inner product. Let $S_1$ and $S_2$ subspaces of $\mathbb{R}^m$ , $P_1\in\mathbb{M}_m(\mathbb{R})$ a orthogonal projection matrix on subspace $S_1$ ...
1
vote
0answers
26 views

Orthogonal projection matrix proof

Let $P\in \mathbb{M}_m(\mathbb{R})$ a orthogonal projection matrix. Show that the matrix $Q=I-P$ is a orthogonal projection matrix. Make a geometric interpretation of the elements $z=Pb$ and ...
0
votes
1answer
18 views

Gaussian elimination problem

$$x_1 + 10x_2 − 3x_3 = 8$$ $$x_1 + 10x_2 + 2x_3 = 13$$ $$x_1 + 4x_2 + 2x_3 = 7$$ when making 2nd and 3rd 1st columns 0 using Gaussian elimination, the second row second column also becomes zero, so ...
0
votes
1answer
15 views

Upperbound on the following logarithmic function with matrix

I am trying to upperbound the expression below with a function $f$ that is a function of the identity matrix $$\log(1+\mathbf{h}^* \mathbf{\Sigma} \mathbf{h}) \leq f( {\bf I},{\bf h })$$ $$\Sigma ...
1
vote
2answers
13 views

sum of matrices with unique solutions

Let $K$ be any field with a characteristic, different than 2, and $A$ any $n \times n$-matrix over $K$. For the equation $A = B + C$, where $A$ and $B$ are $n \times n$-matrices over $K$, are $B = ...
0
votes
3answers
31 views

Find the complex eigenvectors, knowing the eigenvalues

If $$A= \begin{pmatrix}1 & -1 \\ h^2 & 1\end{pmatrix},$$ I know the complex eigenvalues are $1+ih$ and $1-ih$. How do we find the complex eigenvectors? Can someone please explicitly show me ...
0
votes
1answer
35 views

Properties of a matrix that shares the set of real eigenvalues with its inverse

For a $3\times 3$ real matrix, let $c(A)$ denotes the set of real eigenvalues of $A$. Suppose $c(B)=c(B^{-1})$ for a non-singular matrix $B$ with no repeated eigenvalues. Then which of the following ...
1
vote
0answers
29 views

Find the matrix $P$

$A= \begin{bmatrix}1 & -2 & 3\\-2 & 6 &-9 \\3 & -9 & 4 \end{bmatrix}$ Find $P$ with non-negative integer entries and has determinant $2$. $P^TAP=\begin{bmatrix}a & 0 ...
1
vote
2answers
33 views

linear algebra-norm of matrix

Why $ \|A\| = \|A^*\| $ in matrix ? Suppose that A is a normal matrix. I know $ A^* = A^{-1} \det(A) $ and so $\|A^*\| = \|\det(A) A^{-1} \| \rightarrow \|A^*\|=\det(A) \|A^{-1}\|$ but I can't prove ...
1
vote
0answers
26 views

Generate a random neutrally stable matrix

I need to generate random real matrices such that all eigenvalues have real part equal to 0 -- i.e. random neutrally stable matrices. What's the simplest way to do this? Note that I don't care about ...
1
vote
0answers
24 views

Eigenvalue multiplicity of a product of two real skew-symmetric matrices

All the roots of characteristics polynomial of $AB$, where $A$, $B$ are skew symmetric matrices of order $2n$, are of multiplicity greater then $1$. I know that eigen values of skew symmetric ...
9
votes
3answers
152 views

If $\,A^3-A+I=0,\,$ then $A$ is invertible

Prove or disprove. If $A$ is a square matrix and $A^3-A+I=0,$ then $A$ is invertible. Is it possible to say the characteristic polynomial of $A$ is $\,p(t)=t^3-t+1$, and $A$ is invertible since $0$ ...
0
votes
2answers
48 views

Is the set $SL(2, \mathbb F)$ an Abelian group?

For the set $SL(2,\mathbb F)$, where $\mathbb F$ are entries from either $$\mathbb{Q},\mathbb{R},\mathbb{C} \text{ or } \mathbb{Z}_p \text{ (p is prime)}$$ How should I start by checking this matrix ...
0
votes
1answer
44 views

Linear transformations and their kernels

Am I correct to assume all of the following are linear transformations? I tested all 3 for the 2 conditions $T(A_1+A_2)$ and $T(kA)$ but I was unsure about if (a) was a linear transformation. The ...
2
votes
2answers
27 views

Solution Space - Linear Algebra

For a matrix:\begin{bmatrix}-1&2&3&-3&6&7\\ 1&-1&-2&2&-5&-6\\ -1&1&2&-1&2&4\\ -2&2&4&-2&4&8\\\end{bmatrix} To solve for ...
0
votes
1answer
42 views

Least squares and pseudo-inverse

Let $b\in \mathbb{R}^m$,$A\in M_{m\times n}(\mathbb{R})$ with $m>n$ and $rank(A)=n$, and the element $x^*\in \mathbb{R}^m$ solution of least squares of $Ax=b$. i) Show that $r^*=b-Ax^*\in N(A^T)$ ...
0
votes
1answer
25 views

Projection on the coordinate plane

Consider the vector space $\mathbb{R}^3$ with usual inner product. Find the orthogonal projection matrix on coordinate plane $xy$ and $xz$ I think that projection on xy is ...
0
votes
2answers
74 views

Orthogonal projection matrix

Let $A\in M_{m\times n}(\mathbb{R})$. Denoting by $R(A)$ the column space of $A$ and $N(A)$ the null space of $A$. I know that $z^*=Ax^*$ is a projection of $b\in R^m$ on $R(A)=N(A^T)$ where ...
0
votes
1answer
17 views

Effect of spectral shift on the eigenvalues of a real symmetric matrix [duplicate]

Suppose a matrix A(real symmetric) is changed to A − σ I, where σ is any scalar quantity and I is the identity matrix. Explain what happens to the eigenvalues and eigenvectors of A? I am unable to ...
2
votes
1answer
24 views

On the expression of the Galois conjugates in terms of the coordinates in a basis

Let $K$ be a field and let $L$ be a Galois extension of $K$. Assume that $[L:K]=n$, and consider $e=(e_1, e_2, ...,e_n)$ a basis of $L$ over $K$. We note ...
1
vote
0answers
15 views

Confidence interval from covariance matrix

We have a matrix of stochastic variables $X\sim\mathcal{N}(0,\Sigma^2)$, where $\Sigma^2$ is a positive definite covariance matrix. How do we calculate the 95% confidence interval for X? (lets say ...
0
votes
1answer
26 views

Linear Algebra - Change of basis

Let $S$ be the standard basis for $\mathbb{R}^5$. Let $B = (b_1, b_2, b_3, b_4, b_5)$ be the ordered basis with: $b_1 = (2, 1, 1, -2, -2)$; $b_2 = (0, -2, 4, 5, -4)$; $b_3 = (1, -4, 5, 5, -4)$; ...
0
votes
1answer
34 views

Find a matrix whose column space contains the column space of the given matrix.

Let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}\text{.}$$ $C(A)$ denotes the column ...
0
votes
1answer
16 views

Linear functional and Hessian

Consider the vector space $\mathbb{R}^n$ provided with the usual inner product $<.,.>$. Let $A\in \mathbb{M}_n(\mathbb{R})$ a invertible matrix, $b\in\mathbb{R}^n$ and $J:\mathbb{R}^n\rightarrow ...
0
votes
1answer
38 views

Do the spaces spanned by the columns of a matrix and by the columns of a set of matrices coincide?

As in Do the spaces spanned by the columns of the given matrices coincide?, let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ ...
1
vote
1answer
62 views

How to show the following matrix is positive definite?

How to show the following matrix is positive definite. \begin{equation} \sum_{i=1}^n \Big[(d_i^Tp)^2\left\{\left( \begin{array}{c} d_i\\ A_ip \end{array} \right) \left( \begin{array}{c} ...
1
vote
3answers
24 views

Matrix Multiplication || Linear Transformation

This seems to be trivial question but unfortunately I can't figure it out: Here $B,V$ and $U$ are matrices: Do relation $B(V + U)B^{-1} = BVB^{-1} + BUB^{-1}$ hold true? If yes than which matrix ...
1
vote
1answer
40 views

minimal polynomial of linear transformation

Let V and W are finite dimensional vector space over R.$ T_1:V\to V$ and $ T_2:W\to W$ be linear transformation whose minimal polynomials are given by $ f_1(x)=x^3+x^2+x+1 , f_2(x)=x^4-x^2-2$. Let $ ...
2
votes
1answer
20 views

convert the inverse of sum of two hermitian matrices into sum of two or more matrices.

I want to convert the inverse of sum of two hermitian matrices into sum of two or more matrices. I mean I want to simplify the bellow equation in a way that not to have inverse of sum of matrices any ...
0
votes
1answer
21 views

Check for basis of a matrix

Given the matrices in $M_{3,3}$. ...
2
votes
2answers
31 views

Do the spaces spanned by the columns of the given matrices coincide?

Reviewing linear algebra here. Let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \qquad ...
1
vote
0answers
16 views

Time complexity for the multiplication of three rectangular matrix

For the multiplication of two matrix, one can use the classical methods, such as strassen algorithm, to reduce the time complexity. Here, I just wonder if there is any efficent algorithm aiming at the ...
2
votes
2answers
45 views

Solution to $A = BX + YC$ where $A$ is a square matrix of rank $n$, $B,C$ known, rank $m<n$

I hope this isn't too trivial of a problem. I'm really struggling with it and I feel like it shouldn't be that difficult. As stated in the title: Given (full rank): $A\in\mathbb{R}_{nxn}$ ...
0
votes
0answers
18 views

Norm of operator matrix

I'm having trouble with the following: suppose H is a Hilbert space and $f_{i, j}, g_{i, j} : H \rightarrow H$, $1 \leq i, j \leq n$ are bounded operators. Then we have operators $(f_{i, j}) , (g_{i, ...
0
votes
1answer
25 views

determinants of large and infinite matrices

Given a square n x n matrix A, is it possible to find the determinant of the matrix for large values of n easily, and thereby as n goes to infinity? I know that the number of components of the ...
0
votes
0answers
11 views

what's the potential application of low rank approximation of stochastic matrices

Suppose we have a stochastic matrix $P$ for a Markov chain, and we can compute a low rank approximation of $P$, say $P_k$, or we can find the nonnegative matrix factorization of $P$, i.e., $P=AW$ ...
0
votes
1answer
19 views

How do you find the 4x4 matrix corresponding to the transformation T with respect to the basis?

If the transformation $T$ acting on the vector space $A \in Mat_{2,2}$ is given by $T(A)=CA$, where $ C= \left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right) $ how would you find the ...
1
vote
2answers
36 views

Evaluate determinant of an $n \times n$ matrix, help

I need help with this problem: $D_{n}= \begin{vmatrix} 1 & 1 & 0 & \cdots & 0 & 0 & 0 \\ 1 & 1 & 1 & \cdots & 0 & 0 & 0 \\ 0 ...
1
vote
2answers
19 views

Norm Used in Perturbation Matrix Thoery?

My question is that what is the type of 2-Norm used in Weyl's theorem for relative perturbation? Is that a induced norm, or a entry-wise norm? $\epsilon=\|X^T X-I\|_2$, where relative difference in ...