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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0answers
8 views

Finding a nullspace of a matrix - what should I do after finding equations?

I am given the following matrix $A$ and I need to find a nullspace of this matrix. $$A = \begin{pmatrix} 2&4&12&-6&7 \\ 0&0&2&-3&-4 \\ ...
0
votes
3answers
48 views

Prove that A is invertible if $A^2 - 4A -7I = 0$. [duplicate]

The $2 \times 2$ matrix $A$ satisfies $$A^2 - 4A -7I = 0,$$ where $I$ is the identity matrix. Prove that $A$ is invertible. I'm not sure how to do this. Help would be appreciated.
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votes
1answer
16 views

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there… [duplicate]

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there exists a two-dimensional vector ${w}$ such that Aw = v. Show that ${A}$ is invertible. I'm not sure how to do this.
2
votes
3answers
48 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 ...
-2
votes
0answers
11 views

How to find proper parameter t to diagonalise matrix [on hold]

Find the proper parameter $t$ to diagonalise this $3 \times 3$ matrix. $\left|\begin{array}{ccc} 1 & t & 25 \\ 0 & t & t+1 \\ 0 & 0 & -1 \end{array}\right|$ How do I solve ...
0
votes
0answers
14 views

Can someone please help me to prove if a matrix is non-negative?

Let $\textbf{r}_{1}$ and $\textbf{r}_{2}$ be two symmetric, diagonal dominate, Metlzer matrices. Let $\textbf{F}(m)= (\textbf{I}-e^{m\textbf{r}_1})(\textbf{I}-e^{m(\textbf{r}_1+\textbf{r}_2)})^{-1}$. ...
0
votes
1answer
25 views

Proving that matrix in equation is invertible

The $2 \times 2$ matrix ${A}$ satisfies ${A}^2 - 4 {A} - 7 {I} = {0}$ where ${I}$ is the $2 \times 2$ identity matrix. Prove that ${A}$ is invertible. I have tried to solve it like a quadratic, but ...
-2
votes
1answer
35 views

What is the 5th root of the stated matrix? [on hold]

A $3\times3$ matrix $\left[\begin{array}{ccc}1&2&0\\-1&-2&0\\3&5&1\end{array}\right]$ What should I do ? Many thanks
0
votes
2answers
22 views

Orthogonality and projections

1)Consider the vector space $\mathbb{R}^n$ with usual inner product. And let S the subspace generated by $u\in \mathbb{R}^n,u\neq 0$. Find the orthogonal projection matrix $P$ onto the subspace ...
3
votes
1answer
22 views

How to prove this identity involving characteristic polynomials on both sides?

Suppose $A\in \Bbb C^{m\times n},B\in \Bbb C^{n\times m},m\ge n$, prove: $$\det(\lambda I_m-AB)=\lambda^{m-n}\det(\lambda I_n-BA)$$ I don't want to get into nasty determinant calculation. Instead, I ...
1
vote
1answer
16 views

Finding a matrix by using hermitian

$A=\left[ \begin{array}{ccc} 4 & 0 & 0 \\ 0 & 1 & i \\ 0 & -i ...
0
votes
2answers
30 views

what do eigenvalue & eigenvector of $4\times4$ matrix represent?

What do we get when calculating the eigenvalue and eigenvector of a $4\times4$ matrix? What do those values actually represent?
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vote
2answers
20 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 - a}(A ...
2
votes
5answers
36 views

Proving any vector in $\Bbb R^n$ can be written on the form $x = u + v$

I'm having a hard time understanding the solution of this exercise. The exercise says: Let A be an $n\times n$ matrix so that $$A^2 = A$$ Show that every vector $x$ in $\Bbb R^n$ can be written as ...
1
vote
2answers
27 views

How can you find a matrix given you know its kernel/nullspace?

Suppose we are given that $\phi : \mathbb{R}^4 \rightarrow \mathbb{R}^3$, and also that $\ker\phi$ is the span of $\{\begin{pmatrix} 1 \\ 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \\ 0 \\ 1 ...
0
votes
1answer
58 views

vectors and matrix problem [on hold]

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
37 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
23 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
26 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
19 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
45 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
28 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
38 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
vote
1answer
16 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
19 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
13 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
31 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
29 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
16 views

Upperbound on the following logarithmic function with matrix

I am trying to find an upperbound the expression below with a function $f$ that is a function of the identity matrix $$\log(1+\mathbf{h}^* \mathbf{\Sigma} \mathbf{h}) \leq f( {\bf I},{\bf h })$$ ...
1
vote
2answers
14 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
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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
36 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
27 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
26 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
156 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
45 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
30 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
26 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
79 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
18 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
29 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 ...