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Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

255
votes
26answers
58k views

If $AB = I$ then $BA = I$

If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$. I do not understand anything more than the following. Elementary row operations. Linear ...
596
votes
13answers
113k views

What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
43
votes
8answers
17k views

How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?

Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$ I proved $\operatorname{rank}(AB) \...
4
votes
1answer
4k views

Kernels and reduced row echelon form - explanation

The following text is written in my textbook and I don't really understand it: If $A = (a_{ij}) \in$ Mat$(m x N, F)$ is a matrix in reduced row echelon form with $r$ nonzero rows and pivots in the ...
106
votes
10answers
185k views

Inverse of the sum of matrices

I have two square matrices - $A$ and $B$. $A^{-1}$ is known and I want to calculate $(A+B)^{-1}$. Are there theorems that help with calculating the inverse of the sum of matrices? In general case $B^{-...
34
votes
5answers
10k views

Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?

Let $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials? I have a proof only if $ A$ or $ B $ is invertible. Is it true for all cases?
45
votes
5answers
5k views

Similar matrices and field extensions

Given a field $F$ and a subfield $K$ of $F$. Let $A$, $B$ be $n\times n$ matrices such that all the entries of $A$ and $B$ are in $K$. Is it true that if $A$ is similar to $B$ in $F^{n\times n}$ then ...
47
votes
4answers
36k views

Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$? (Citation needed!)

First of all, am I being crazy in thinking that if $\lambda$ is an eigenvalue of $AB$, where $A$ and $B$ are both $N \times N$ matrices (not necessarily invertible), then $\lambda$ is also an ...
75
votes
1answer
48k views

Simultaneously Diagonalizable Proof

Two $n\times n$ matrices $A, B$ are said to be simultaneously diagonalizable if there is a nonsingular matrix $S$ such that both $S^{-1}AS$ and $S^{-1}BS$ are diagonal matrices. a.) Show that ...
48
votes
9answers
88k views

How to show that $\det(AB) =\det(A) \det(B)$?

Given two square matrices $A$ and $B$, how do you show that $$\det(AB) = \det(A)\det(B)$$ where $\det(\cdot)$ is the determinant of the matrix?
35
votes
3answers
49k views

Is the product of symmetric positive semidefinite matrices positive definite?

I see on Wikipedia that the product of two commuting symmetric positive definite matrices is also positive definite. Does the same result hold for the product of two positive semidefinite matrices? ...
17
votes
3answers
5k views

The Center of $\operatorname{GL}(n,k)$

The given question: Let $k$ be a field and $n \in \mathbb{N}$. Show that the centre of $\operatorname{GL}(n, k)$ is $\lbrace\lambda I\mid λ ∈ k^∗\rbrace$. I have spent a while trying to prove this ...
100
votes
20answers
188k views

Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
23
votes
4answers
7k views

Sylvester's determinant identity

Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then $$ \det(I_m+AB) = \det(I_n+BA)$$ where $I_m$ and $I_n$ denote the $m \times m$ ...
21
votes
6answers
13k views

Determinant of a block lower triangular matrix

I'm trying to prove the following: Let $A$ be a $k\times k$ matrix, let $D$ have size $n\times n$, and $C$ have size $n\times k$. Then, $$\det\left(\begin{array}{cc} A&0\\ C&D \end{array}\...
20
votes
5answers
7k views

The characteristic and minimal polynomial of a companion matrix

The companion matrix of a monic polynomial $f \in F\left[x\right]$ in 1 variable $x$ over a field $F$ plays an important role in understanding the structure of finite dimensional $F[x]$-modules. It ...
30
votes
2answers
13k views

Order of general- and special linear groups over finite fields.

Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have? $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$ Here GL is the general ...
12
votes
3answers
4k views

Proof of this result related to Fibonacci numbers: $\begin{pmatrix}1&1\\1&0\end{pmatrix}^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}$?

$$\begin{pmatrix}1&1\\1&0\end{pmatrix}^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}$$ Somebody has any idea how to go about proving this result? I know a proof by ...
91
votes
14answers
29k views

Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at http://en.wikipedia.org/wiki/Rank_of_a_linear_transformation and I understand the ...
88
votes
7answers
105k views

Show that the determinant of $A$ is equal to the product of its eigenvalues

Show that the determinant of a matrix $A$ is equal to the product of its eigenvalues $\lambda_i$. So I'm having a tough time figuring this one out. I know that I have to work with the characteristic ...
2
votes
4answers
1k views

Given a $4\times 4$ symmetric matrix, is there an efficient way to find its eigenvalues and diagonalize it?

I have a $4\times 4$ matrix $$A=\left(\begin{array}{cccc}8 & 11 & 4 & 3\\11 & 12 & 4 & 7\\4 & 4 & 7 & 12\\3 & 7 & 12 & 17\end{array}\right).$$ I want to ...
76
votes
9answers
119k views

When is matrix multiplication commutative?

I know that matrix multiplication in general is not commutative. So, in general: $A, B \in \mathbb{R}^{n \times n}: A \cdot B \neq B \cdot A$ But for some matrices, this equations holds, e.g. A = ...
15
votes
8answers
5k views

How to calculate the following determinants (all ones, minus $I$)

How do I calculate the determinant of the following $n\times n$ matrices $ \left[ \begin {matrix} 0 & 1 & \ldots & 1 \\ 1 & 0 & \ldots & 1 \\ \vdots & \vdots & \...
25
votes
5answers
45k views

How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$? [duplicate]

How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
12
votes
2answers
2k views

How does the SVD solve the least squares problem?

How do I prove that the least-squares solution for $$\text{minimize} \quad \|Ax-b\|_2$$ is $A^{+} b$, where $A^{+}$ is the pseudoinverse of $A$?
139
votes
14answers
45k views

Intuition behind Matrix Multiplication

If I multiply two numbers, say $3$ and $5$, I know it means add $3$ to itself $5$ times or add $5$ to itself $3$ times. But If I multiply two matrices, what does it mean ? I mean I can't think it ...
31
votes
6answers
14k views

$AB-BA=I$ having no solutions

The following question is from Artin's Algebra. If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions. I have no idea on how to tackle this question. I ...
50
votes
3answers
30k views

Matrices commute if and only if they share a common basis of eigenvectors?

I've come across a paper that mentions the fact that matrices commute if and only if they share a common basis of eigenvectors. Where can I find a proof of this statement?
40
votes
6answers
19k views

why determinant is volume of parallelepiped in any dimensions

for $n = 2,$ I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculate the area by coordinates. But how can one easily realize that it is true ...
16
votes
4answers
22k views

Prove that $A+I$ is invertible if $A$ is nilpotent [duplicate]

Possible Duplicate: Units and Nilpotents Given $A^{2012}=0$ prove that $A+I$ is invertible and find an expression for $(A+I)^{-1}$ in terms of $A$. ($I$ is the identity matrix).
22
votes
4answers
7k views

$I_m - AB$ is invertible if and only if $I_n - BA$ is invertible.

The problem is from Algebra by Artin, Chapter 1, Miscellaneous Problems, Exercise 8. I have been trying for a long time now to solve it but I am unsuccessful. Let $A,B$ be $m \times n$ and $n \...
13
votes
5answers
9k views

How to compute the determinant of a tridiagonal matrix with constant diagonals?

How to show that the determinant of the following $(n\times n)$ matrix $$\begin{pmatrix} 5 & 2 & 0 & 0 & 0 & \cdots & 0 \\ 2 & 5 & 2 & 0 & 0 & \cdots & ...
17
votes
2answers
5k views

Show $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A+B)$

Show $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A+B)$, where $A,B \in M_{m\times n}(\mathbb{F})$. I'm trying to think in terms of linear transformations. We can define $...
7
votes
3answers
4k views

I don't understand why the inverse is this?

my question is related to matrix inverting and Hill cipher(you don't have to know what it is to help me) My teacher gave me an example. First we have a matrix (the key matrix) that multiplied by a ...
9
votes
3answers
4k views

How to show that the nth power of a $n \times n$ nilpotent matrix equals to zero $A^n=0$

$A$ is a $n\times n$ matrix such that $ A^m = 0 $ for some positive integer $m$. Show that $A^n = 0$. My attempt: For $n > m$, it's obvious since matrix multiplication is associative. For $n &...
37
votes
2answers
21k views

Does non-symmetric positive definite matrix have positive eigenvalues?

I found out that there exist positive definite matrices that are non-symmetric, and I know that symmetric positive definite matrices have positive eigenvalues. Does this hold for non-symmetric ...
21
votes
3answers
5k views

Traces of all positive powers of a matrix are zero implies it is nilpotent

Let $A$ be an $n\times n$ complex nilpotent matrix. Then we know that because all eigenvalues of $A$ must be $0$, it follows that $\text{tr}(A^n)=0$ for all positive integers $n$. What I would like ...
26
votes
2answers
3k views

Why does the inverse of the Hilbert matrix have integer entries?

Let $A$ be the $n\times n$ matrix given by $$A_{ij}=\frac{1}{i + j - 1}$$ Show that $A$ is invertible and that the inverse has integer entries. I was able to show that $A$ is invertible. ...
16
votes
2answers
7k views

Why does the $n$-th power of a Jordan matrix involve the binomial coefficient?

I've searched a lot for a simple explanation of this. Given a Jordan block $J_k(\lambda)$, its $n$-th power is: $$ J_k(\lambda)^n = \begin{bmatrix} \lambda^n & \binom{n}{1}\lambda^{n-1} & \...
50
votes
5answers
49k views

Is the rank of a matrix the same of its transpose? If yes, how can I prove it?

I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view: "The rank of a matrix A is the number of non-zero ...
53
votes
3answers
79k views

A matrix and its transpose have the same set of eigenvalues

Let $ \sigma(A)$ be the set of all eigenvalues of $A$. Show that $ \sigma(A) = \sigma(A^T)$ where $A^T$ is the transposed matrix of $A$.
37
votes
3answers
21k views

Eigenvalues of the rank one matrix $uv^T$

Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in ${\mathbb R}^n$, $n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of ...
127
votes
2answers
29k views

What is the geometric interpretation of the transpose?

I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the ...
40
votes
2answers
21k views

Matrix multiplication: interpreting and understanding the process

I have just watched the first half of the 3rd lecture of Gilbert Strang on the open course ware with link: http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/ It ...
52
votes
5answers
56k views

How do I tell if matrices are similar?

I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not. I solved this by using a matrix called $S$: $$\left(\begin{array}{cc} a& b\\ ...
20
votes
3answers
6k views

Why do the $n \times n$ non-singular matrices form an “open” set?

Why is the set of $n\times n$ real, non-singular matrices an  open subset of the set of all $n\times n$ real matrices? I don't quite understand what "open" means in this context. Thank you.
22
votes
1answer
5k views

Probability that a random binary matrix is invertible?

What is the probability that a random $\{0,1\}$, $n \times n$ matrix is invertible? Assume the 0 and 1 are each present in an entry with probability $\frac{1}{2}$. Is there an explicit formula as a ...
8
votes
3answers
24k views

A rank-one matrix is the product of two vectors

Let $A$ be an $n\times m$ matrix. Prove that $\operatorname{rank} (A) = 1$ if and only if there exist column vectors $v \in \mathbb{R}^n$ and $w \in \mathbb{R}^m$ such that $A=vw^t$. Progress: I'm ...
83
votes
3answers
17k views

Why, historically, do we multiply matrices as we do?

Multiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very ...
33
votes
6answers
19k views

Sylvester rank inequality: $\operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n$

If $A$ and $B$ are two matrices of the same order $n$, then $$ \operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n. $$ I don't know how to start proving this inequality. I ...