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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.

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26
votes
1answer
76k views

What does double vertical-line means in linear algebra?

I have a formula, which I have no idea how to solve, because I don't know that double vertical-line sign: $\|{\rm Ax} \|$? $${\rm x} \ne 0 \in \Bbb R^n, \quad 0 < m \le \frac {\| {\rm Ax} \|} {\| {...
26
votes
5answers
152k views

Diagonalizable Matrices: How to determine?

I am trying to figure out how to determine the diagonalizability of the following two matrices. For the first matrix $$\left[\begin{matrix} 0 & 1 & 0\\0 & 0 & 1\\2 & -5 & 4\...
26
votes
3answers
4k views

Eigenvalues of $A$ and $A + A^T$

This question has popped up at me several times in my research in differential equations and other areas: Let $A$ be a real $N \times N$ matrix. How are the eigenvalues of $A$ and $A + A^T$ related? ...
26
votes
1answer
24k views

block matrix multiplication

If $A,B$ are $2 \times 2$ matrices of real or complex numbers, then $$AB = \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right]\cdot \left[ \begin{array}{cc} b_{11} ...
25
votes
5answers
3k views

Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that $A^2$ is diagonal. Must $A$ be diagonal?

Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that $A^2$ is diagonal? Must $A$ be diagonal. In other words, is it true that $$A^{2}\;\text{is diagonal}\;\Longrightarrow a_{ij}=0,\;i\neq j\;\;?...
25
votes
6answers
5k views

Why is the Operator Norm so hard to calculate?

I recently took a better look at the operator norm defined on a matrix $\mathbf A \in \Bbb{K}^{n\times n}$ as follows: $$ \|\mathbf A\|_p=\sup\{\|\mathbf Ax\|_p \mid x\in\Bbb{K}^n\land\|x\|=1\} $$ ...
25
votes
7answers
14k 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}\...
25
votes
2answers
3k views

Why is the norm of a matrix larger than its eigenvalue?

I know there are different definitions of Matrix Norm, but I want to use the definition on WolframMathWorld, and Wikipedia also gives a similar definition. The definition states as below: Given a ...
25
votes
4answers
30k views

Proving: “The trace of an idempotent matrix equals the rank of the matrix”

How could we prove that the "The trace of an idempotent matrix equals the rank of the matrix"? This is another property that is used in my module without any proof, could anybody tell me how to prove ...
25
votes
2answers
22k views

Norm of a symmetric matrix equals spectral radius

How do I prove that the norm of a matrix equals the absolutely largest eigenvalue of the matrix? This is the precise question: Let $A$ be a symmetric $n \times n$ matrix. Consider $A$ as an ...
25
votes
3answers
2k views

Checkboard matrix, brand new or old?

Ok so what I found was a square matrix of order $n×n$ where $n$ follows $2m+1$ and $m$ is a natural number the pattern these matrices follow is as follows: for a $3×3$ matrix: $$ A = \left( \begin{...
25
votes
1answer
12k views

sum of squares of dependent gaussian random variables

Ok, so the Chi-Squared distribution with n degrees of freedom is the sum of the squares of n independent Gaussian random variables. The trouble is, my Gaussian random variables are not independent. ...
25
votes
2answers
64k views

Understanding rotation matrices

How does $ {\sqrt 2 \over 2} = \cos (45^\circ)$? Is my graph (the one underneath the original) accurate with how I've depicted the representation of the triangle that the trig function represent? ...
25
votes
3answers
30k views

Matrices: left inverse is also right inverse? [duplicate]

If $A$ and $B$ are square matrices, and $AB=I$, then I think it is also true that $BA=I$. In fact, this Wikipedia page says that this "follows from the theory of matrices". I assume there's a nice ...
25
votes
3answers
1k views

Sudoku with special properties

Sudoku is a puzzle, with the objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid (also "sudoku-blocks") contains all of ...
25
votes
1answer
852 views

A surprising result about the product of Blaschke matrices

I have verified analytically the conjecture described bellow up to $n=4$, but have had no success trying to prove it. Any help would be much appreciated. Setup Let $\{\lambda_i\}_{i=1}^n$ be real ...
24
votes
5answers
3k views

If $A$ and $B$ are matrices such that $AB^2=BA$ and $A^4=I$ then find $B^{16}$.

If $A$ and $B$ are matrices such that $AB^2=BA$ and $A^4=I$, then find $B^{16}$. My Method: Given $$AB^2=BA \tag{1}$$ Post multiplying with $B^2$ we get $$AB^4=BAB^2=B^2A$$ Hence $$AB^4=B^2A$$ Pre ...
24
votes
3answers
2k views

Are $10\times 10$ matrices spanned by powers of a single matrix?

I don't know how to answer this question: Is there a $10 \times 10$ matrix $A$ such that $$M_{10}(\mathbb{F})=\text{span}\{I,A,A^2,\ldots, A^{100}\}\textrm{,}$$ where $M_{10}(\mathbb{F})$ is the ...
24
votes
4answers
2k views

Suppose $M$ is an $m \times n$ matrix such that all rows and columns of $M$ sum to $1$. Show that $m=n$

I find this a rather awkward question, and I was given a hint: use invariants, which I found even more awkward. Suppose $M$ is an $m \times n$ matrix such that all rows and columns of $M$ sum to $1$...
24
votes
5answers
3k views

What kind of matrix is this and why does this happen?

So I was studying Markov chains and I came across this matrix \begin{align*}P=\left( \begin{array}{ccccc} 0 & \frac{1}{4} & \frac{3}{4} & 0 & 0\\ \frac{1}{4} & 0 &...
24
votes
4answers
8k 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 $m \times m$ ...
24
votes
3answers
37k views

Inverse of a Positive Definite

Let K be nonsingular symmetric matrix, prove that if K is a positive definite so is $K^{-1}$ . My attempt: I have that $K = K^T$ so $x^TKx = x^TK^Tx = (xK)^Tx = (xIK)^Tx$ and then I don't know ...
24
votes
8answers
3k views

What is the theory of Matrices?

I've just passed high school and studying matrices. I've learned about determinant, transpose and adjoint etc. I've learned the method of finding these things but what's the purpose of finding these ...
24
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 ...
24
votes
4answers
2k views

What is an interpretation of the matrix exponential?

I just read about the existence of the "matrix exponential" $$e^X := \sum_{k = 0}^\infty\frac1{k!}X^k$$ Is there a simple way to interpret this? I understand the analog between real number ...
24
votes
1answer
27k views

Elegant proofs that similar matrices have the same characteristic polynomial?

It's a simple exercise to show that two similar matrices has the same eigenvalues and eigenvectors (my favorite way is noting that they represent the same linear transformation in different bases). ...
24
votes
5answers
37k views

The transpose of a permutation matrix is its inverse.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...
24
votes
2answers
498 views

For each $a \in \mathbb{R}$ evaluate $ \lim\limits_{n \to \infty}\left(\begin{smallmatrix}1&\frac{a}{n}\\\frac{-a}{n}&1\end{smallmatrix}\right)^n$

If $a \in \mathbb{R}$, evaluate $$ \lim_{n \to \infty}\left(\begin{matrix} 1&\frac{a}{n}\\\frac{-a}{n}&1\end{matrix}\right)^{n}$$ My attempt: Let $$A = \left(\begin{matrix} 0&a\\-a&0\...
24
votes
5answers
3k views

Why do we need a Jordan normal form? [duplicate]

My professor said that the main idea of finding a Jordan normal form is to find the closest 'diagonal' matrix that is similar to a given matrix that does not have a similar matrix that is diagonal. I ...
24
votes
6answers
19k views

Matrix Inverses and Eigenvalues

I was working on this problem here below, but seem to not know a precise or clean way to show the proof to this question below. I had about a few ways of doing it, but the statements/operations were ...
24
votes
3answers
17k views

Sum of positive definite matrices still positive definite?

I have two matrices, which are square, symmetric, and positive definite. I would like to prove that the sum of the two matrices still have the same properties, that is square, symmetric, and positive ...
23
votes
6answers
4k views

Is every noninvertible matrix a zero divisor?

Is every noninvertible matrix over a field a zero divisor? Related to this: What are sufficient conditions for a matrix to be a zero divisor over a noncommutative ring?
23
votes
2answers
43k views

Expressing the determinant of a sum of two matrices?

Can $$\det(A + B)$$ be expressed in terms of $$\det(A), \det(B), n$$ where $A,B$ are $n\times n$ matrices? # I made the edit to allow $n$ to be factored in.
23
votes
4answers
2k views

How to divide by a matrix

I found a question in an old exam, where the function $\phi(z) := \frac{\exp(z) - 1}{z}$ is given. Now we evaluate $\phi(\mathbf{A})$. But how do I divide by a matrix? I already thought about ...
23
votes
5answers
15k views

Symmetric matrix is always diagonalizable?

I'm reading my linear algebra textbook and there are two sentences that make me confused. (1) Symmetric matrix $A$ can be factored into $A=Q\lambda Q^{T}$ where $Q$ is orthogonal matrix : ...
23
votes
4answers
55k views

Eigenvalues in orthogonal matrices

Let $A \in M_n(\Bbb R)$. How can I prove, that 1) if $ \forall {b \in \Bbb R^n}, b^{t}Ab>0$, then all eigenvalues $>0$. 2) if $A$ is orthogonal, then all eigenvalues are equal to $-1$ or $...
23
votes
3answers
692 views

Multiplying by a $1\times 1$ matrix?

For matrix multiplication to work, you have to multiply an $m \times n$ matrix by an $n \times p$ matrix, so we have $$\bigg(m \times n\bigg)\bigg( n\times p \bigg).$$ But what about a $1 \times 1$ ...
23
votes
5answers
2k views

Is there anything special about this matrix?

I've just encountered a matrix which seems to display nothing special to me: $$B=\begin{pmatrix}1&4&2\\0 &-3 &-2\\ 0 &4 &3 \end{pmatrix}$$ But further observation reveals ...
23
votes
5answers
4k views

Let $A$ and $B$ be $n \times n$ real matrices such that $AB=BA=0$ and $A+B$ is invertible

I came across the following problem that says: Let $A$ and $B$ be $n \times n$ real matrices such that $AB=BA=0$ and $A+B$ is invertible. Then how can I prove the following: rank $A$+ ...
23
votes
2answers
12k views

Are one-by-one matrices equivalent to scalars?

I am a programmer, so to me $[x] \neq x$—a scalar in some sort of container is not equal to the scalar. However, I just read in a math book that for $1 \times 1$ matrices, the brackets are often ...
23
votes
2answers
24k views

What does Determinant of Covariance Matrix give?

I am representing my 3d data in convariance matrix. I just want to know what the determinant of Convariance Matrix gives. If the determinant is positive, zero, negative, high positive, high negative. ...
23
votes
2answers
9k views

Topology of matrices

1.Consider the set of all $n×n$ matrices with real entries as the space $\mathbb R^{n^2}$ . Which of the following sets are compact? (a) The set of all orthogonal matrices. (b) The set of all ...
23
votes
2answers
546 views

Showing $\prod\limits_{i<j} \frac{x_i-x_j}{i-j}$ is an integer

Let $x_1,...,x_n$ be distinct integers. Prove that $$\prod_{i<j} \frac{x_i-x_j}{i-j}\in \mathbb Z$$ I know there is a solution using determinant of a matrix, but I can't remember it now. Any ...
23
votes
1answer
22k views

Prove: symmetric positive definite matrix

I'm studying for my exam of linear algebra.. I want to prove the following corollary: If $A$ is a symmetric positive definite matrix then each entry $a_{ii}> 0$, ie all the elements of the ...
23
votes
1answer
6k 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 ...
23
votes
1answer
1k views

How to find eigenvalues and eigenvectors of this matrix

Can you help to find eigenvalues and eigenvectors of the following matrix? Here is the matrix: $$C = \small \begin{pmatrix} -\sin(\theta_{2} - \theta_{M}) & \sin(\theta_{1} - \theta_{M}) & 0 ...
23
votes
0answers
818 views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using this formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $$V= \begin{bmatrix} ...
22
votes
6answers
4k views

Compute the $n$-th power of triangular $3\times3$ matrix

I have the following matrix $$ \begin{bmatrix} 1 & 2 & 3\\ 0 & 1 & 2\\ 0 & 0 & 1 \end{bmatrix} $$ and I am asked to compute its $n$-th power (to express each element as a ...
22
votes
6answers
5k views

How to show path-connectedness of $GL(n,\mathbb{C})$

Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to ...
22
votes
6answers
3k views

What is the most rigorous definition of a matrix?

Is matrix just a rectangular array of symbols and expressions, or one can define it in a more formal way?