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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1answer
33 views

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$?

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$? I read a book a very long time ago where it said something like this: Given a system of two equations with two ...
0
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1answer
19 views

Using Gauss elimination to check for linear dependence

I have been trying to establish if certain vectors are linearly dependent and have become confused (in many ways). when inputting the vectors into my augmented matrix should they be done as columns or ...
0
votes
3answers
25 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
54 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.
0
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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.
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3answers
52 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
17 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 $\textbf{F}(m)= (\textbf{I}-e^{m\textbf{r}_1})(\textbf{I}-e^{m(\textbf{r}_1+\textbf{r}_2)})^{-1}$. ...
1
vote
1answer
53 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 ...
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votes
1answer
36 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
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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
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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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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
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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 ...
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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
27 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
39 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
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1answer
17 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
32 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
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0answers
30 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
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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
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
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
80 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
30 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: $b_1 = (2, 1, 1, -2, -2)$; $b_2 = (0, -2, 4, 5, -4)$; $b_3 = (1, -4, 5, 5, -4)$; ...