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
349 views

Calculating angle of rotation of orthogonal 3x3 matrix

Regarding the matrix in Q3b here: http://www.maths.ox.ac.uk/system/files/coursematerial/2013/2637/5/13sh2.pdf I've worked out the axis of rotation by finding out the line of invariant points, but I'm ...
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49 views

Given $A$, $B$ and $C=(I+AB)^{-1}$ find $(I-BCA)(I+BA)$

All matrices are square and $(I+AB)$ is invertible. Part B of the exercise is to prove that $(I+BA)$ is invertible. Quite frankly I have no idea how to do this, I've tried rearanging the matrices but ...
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1answer
43 views

$A,B\in\mathbb R^{n\times n}$ then $ \|A^{-1}-B^{-1}\|\leq\|A^{-1}\|\|B^{-1}\|\|A-B\|$

I am reworking my lectures and in one proof our prof used the following: Let $A,B\in\mathbb R^{n\times n}$ invertible. Then $ \|A^{-1}-B^{-1}\|\leq\|A^{-1}\|\|B^{-1}\|\|A-B\|$ Unfortunately I have ...
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1answer
60 views

Is this Jordan decomposition possible?

Is this Jordan form possible? $$J=\begin{pmatrix} \lambda & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & \lambda & 0 & 0 & 0 & 0 & 0\\ 0 ...
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133 views

Diaonalized Matrix of the form $S^2=D$

If $D$ is a diagonal matrix, with non-negative eigenvalues, prove that there is a matrix $S$ such that $S^2 = D$
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2answers
28 views

Prove that matrix with parameter is positive definite

I want to prove that the following matrix is positive definite for $a \in (0.5,1)$. \begin{align} A = \begin{bmatrix} 1 & a & a \\ a & 1 & a \\ a & a & 1 \end{bmatrix} ...
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53 views

Questions regarding minors of a positive definite matrix

In my lectures on Matrix computations, there's a section titled Gaussian elimination and Cholesky decomposition. It is a follows: suppose $A=A^T$ is positive definite, $a_{11>0}$ and $A_{22}$ is ...
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100 views

Question regarding 3 x 3 matrices

If $A$ is a $3 \times 3$ matrix with real elements and $\det(A)=1$, then are these affirmations equivalent: $$ \det(A^2-A+I_3)=0 \leftrightarrow \det(A+I_3)=6 \text{ and } \det(A-I_3)=0? $$
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1answer
781 views

Why does spectral norm equal the largest singular value?

This may be a trivial question yet I was unable to find an answer: $$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$ where the spectral norm $\left \| A \right ...
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1answer
221 views

How prove this matrix equation $AXB=C$ has a solution

Qustion: if Matrix equation $AY = C$ and $ZB = C$ has solution, show that: the equation of $AXB= C$ has solution This problem is from this PDF(page 3) problem 2 ...
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74 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 ...
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2answers
29 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 ...
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3answers
154 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.
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1answer
670 views

Prove that every unitary matrix U is unitarily diagonalizable.

I just can't show that a unitary matrix U is unitarily diagonizable. I know I need to show that U is unitarily similar to a diagonal matrix, and this result is presumably a consequence of the spectral ...
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2answers
2k views

Inverse of a 4x4 matrix with variables

I missed my class on the inverses of matrices. I'm catching up well, but there's a problem in the book that got me stumped. It's a 4x4 matrix that is almost an identity matrix, but the bottom row ...
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1answer
40 views

$\det(B\cdot A\cdot B^T)\neq0$ if and only if $\ker(B^T)=\{\bar{0}\}$

If we have: $A$, $n\times n$ matrix non singular. $B$, $m\times n$ matrix. How would we prove that $\det(B\cdot A\cdot B^T)\neq0$ if and only if $\ker(B^T)=\{\bar{0}\}$.
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1answer
215 views

Nilpotent matrix and Jordan form

Could you help me solve this problem? Give an example of two nilpotent matrices $N_1$, $N_2$ $ \in M_{n,n} (\mathrm{F})$ with $N_1N_2 = N_2N_1$ such that there is no matrix B with $B^{-1}N_1B$ and ...
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1answer
1k views

Deducing that a matrix is indefinite using only its leading principal minors

$A$ is indefinite iff $A$ fits none of the above criteria. Equivalently, $A$ has both positive and negative eigenvalues. Also equivalently, $x^TAx$ is positive for at least one $x$ and ...
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2answers
114 views

Complex matrices and sets question

I'm having some problems answering a question set for my undergrad maths course. The question is: Find the set $S=\left\{(x,y)\in\mathbb C\times\mathbb ...
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2answers
52 views

Spot my error in solving a linear system

I almost always get the unit matrix if I try to get to an row reduced echelon form. I probably always make a mistake. Can you spot the error? What illegal operations could a beginner do while trying ...
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1answer
36 views

Uniqueness of map by dot product

I know that for a map on a complex vector space we have that if $\langle Ax,x \rangle = 0$ then $A = 0$ via the standard polarization trick. But what is the case if we are talking about real vector ...
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1answer
101 views

Show that $\det(A-\lambda B)$ is a nonconstant polynomial if $B$ is invertible

Let $A$ and $B$ be arbitrary complex square matrices. If $B$ is invertible, show that $$p(\lambda)=\det(A-\lambda B)$$ is a nonconstant polynomial in $\lambda$.
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661 views

product of m x n matrix with n x m matrix

How to prove that product of $\mathbb{m x n}$ matrix with $\mathbb{n x m}$ matrix is not invertible given $\mathbb{m >n}$. For the case of $\mathbb{2 x 1}$ and $\mathbb{1 x 2}$ matrix, it is ...
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1answer
111 views

Conjugation by elementary matrices

Let $ A=\begin{pmatrix} a & b \\ c & d \end{pmatrix} $ be a real matrix, with $c$ not zero. Show that using conjugation by elementary matrices, one can "eliminate" the $a$ entry.
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3answers
71 views

Expressing a $3\times 3$ determinant as the product of four factors

I am attempting to express the determinant below as a product of four linear factors $$\begin{vmatrix} a & bc & b+c\\ b & ca & c+a\\ c & ab & a+b\\ \end{vmatrix} = ...
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2answers
307 views

LU decomposition by hand

Can someone show me a step by step solution to calculate the $LU$ decompisition of the following matrix: $A = \begin{bmatrix} 5 & 5 & 10 \\ 2 & 8 & 6 \\ 3 & 6 ...
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1answer
41 views

Determining order of matrices in $GL_2(\mathbb{F}_7)$

I need to determine the orders of the following matrices in the group $GL_2(\mathbb{F}_7)$: $$\begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix} \text{ and } \begin{pmatrix} 2 & 0\\ 0 & 1 ...
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1answer
2k views

Calculating the number of operations in matrix multiplication

Is there a formula to calculate the number of multiplications that take place when multiplying 2 matrices? For example $$\begin{pmatrix}1&2\\3&4\end{pmatrix} \times ...
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2answers
302 views

How to find out that a matrix is positive definite?

Since somebody here told me that it is in general insufficient to show that a matrix is positive definite when all eigenvalues are positive. I am interested in finding good ways to prove this. In ...
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4answers
127 views

Calculate the determinant of the matrix $(a_{ij})$ where $a_{ij}=a+b$ when $i=j$, and $a_{ij}=a$ otherwise

The matrix is $n\times n$ , defined as the following: $$ a_{ij}=\begin{cases} a+b & \text{ when } i=j,\\ a & \text{ when } i \ne j \end{cases}. $$ When I calculated it I got the ...
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1answer
134 views

Prove that $A^TD-C^TB=I$

Let A,B,C,D be complex matrices $n \times n$ such that $AB^T,CD^T$ are symmetric and $AD^T-BC^T=I$. Prove that $A^TD-C^TB=I$. Can anyone give me any idea? Thank you.
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2answers
115 views

Presentation of finite rings (fields)

One knows that every finite group is isomorphic to a subgroup of $\operatorname{GL}(n)$ for some $n$ large enough. Can every finite ring be represented by a ring of matrices, i.e., is every ring ...
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2answers
204 views

matrix representation of polynomial

Here is a polynomial $p(x,y) = (ax + by)^2$, it can be written like this $$p(x,y) = \left(\left[ \begin{array}{cc} a & b \\ \end{array} \right] \left[ \begin{array}{c} x\\ y\\ \end{array} ...
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2answers
175 views

Let $A$ be a complex $3 \times 3$ matrix with $A^3 = -I$. [duplicate]

Let $A$ be a complex $3 \times 3$ matrix with $A^3 = -I$. Which of the following statements are correct? $A$ has three distinct eigenvalues. $A$ is diagonalizable over $\mathbb{C}$. $A$ ls ...
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2answers
358 views

Determinant of a Matrix Proof: $\;\det(qA) = q^n(\det A)$ [duplicate]

I am required to show that: $\det(qA) = q^n(\det A)$, where $A$ is a real $n\times n$ Matrix, and $q$ is a constant I believe that this claim is true after doing few examples. However, but I do not ...
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1answer
89 views

How to construct a non-diagonalizable matrix with a particular set of eigenvalues

Given a set of eigenvalues, how would you go about constructing a matrix with those particular eigenvalues? I know that you can construct a diagonalizable matrix with those eigenvalues using a ...
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3answers
94 views

Solve $(A^TA)^{−1}(X+B^T)(C^{−1}B^{−1})^T =I$

Let $A, B, C \in M_n(\Bbb R)$ be invertible. Find $X \in M_n(\Bbb R)$ such that $(A^TA)^{−1}(X+B^T)(C^{−1}B^{−1})^T =I$. Express $X$ using no inverses and at most two transposes and show your choice ...
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94 views

Let $A$ and $B$ be $n \times n$ complex matrices. Pick out the true statements.

Let $A$ and $B$ be $ n \times n$ complex matrices. Pick out the true statements: a) If $A$ and $B$ are diagonalizable, so is $A + B$ b) If $A$ and $B$ are diagonalizable, so is $AB$ ...
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4answers
113 views

Prove that the rank of $(1-I)$ is $n$

The rank of $(1-I_n)$, where $1$ is the $n \times n$ all-1 matrix and $I_n$ the $n \times n$ identity matrix, seems to be $n$. How to prove this concisely?
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1answer
2k views

Projection matrix onto null space

I have a matrix H and I want to find the projection matrix onto null space. How can I do this? Sorry if my question seems naive. Thank you, Tanja
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3answers
103 views

Determinant of a matrix

Having some problems with a determinant of a 4x4 matrix M. $ M = \left( {\begin{array}{cc} 1 & 2 & 3 &-1 \\ 0 & 1 & 2 & 2 \\ 1 &1 &0 &0 \\ 3&1&2&0 ...
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1answer
46 views

Nonsingularity of a block matrix

Let $X=\left(\begin{array}{cc} A & B\\ C & 0 \end{array}\right)$ and: If $X$ is non-singular, is $A$ non-singular when $B$ is full column rank and $C$ is full row rank?
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147 views

Eigenvalues of a block matrix

For $X=\left(\begin{array}{cc} A & B\\ C & 0 \end{array}\right)$ , how are eigenvalues of X related to eigenvalues of A ?
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1answer
1k views

What is matrix reduction to normal form PAQ?

Here is my university syllabus. I started doing math in vacation just to get a head start because I am a dunce in math. So, I began with chapter 2 - matrices - because it looked easier. I went half ...
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1answer
773 views

Find non-singular matrices P and Q such that PAQ is in the normal form for the matrix A.

$A= \left[ \begin{array}{ccc} 1 & 2 & 3 & -2 \\ 2 & -2 & 1 & 3 \\ 3 & 0 & 4 & 1 \end{array} \right]$ $A=IAI$ $\left[ \begin{array}{ccc} 1 & 2 & 3 & -2 ...
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2answers
151 views

Matrix equation involving a Pauli matrix

I should solve the following problem: find the matrix $A$ that satisfies the following equation: $$\sin(\pi A)+\cos(\pi A)^2= \left( \begin{array}{ccc} 0 & 1 \\ 1 & 0 \end{array} \right)$$ How ...
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1answer
652 views

Cramer's Rule, 2x2 Matrix

Solve the following system using Cramer's Rule. $$2x + y = 1$$ $$x - 4y = 14$$ I haven't done Cramer's rule for 2x2 matrices, but I figured that the same rules applied as in a 3x3, here's what I ...
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1answer
59 views

Simplifying an equality of products of matrices

I have five matrices $A$, $B$, $C$, $D$, $E$, and I know that $ABC = DEC$. Can I conclude that $AB = DE$?
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1answer
47 views

$(P\Lambda P^{-1}=T^2)~\implies~(\exists \Lambda'~\text{s.t.}~T=R\Lambda' R^{-1})$: $\;P,R\;$ Unitary Matrices

Let $T$ be a linear operator such that the operator $T^2$ is diagonalizable. Is $T$ necessarily diagonalizable?
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2answers
469 views

Finding invariant factors of finitely generated Abelian group

There is this question that I wasn't sure how to do but somehow got the answers partially correct (maybe). Suppose that the abelian group $M$ is generated by three elements $x,y,z$ subject to the ...