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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.

A matrix is a rectangular array of elements, usually numbers or variables, arranged in rows and columns. A matrix with $m$ rows and $n$ columns has $m \times n$ elements and is called an $m$ by $n$ matrix. Matrices are a part of .

Matrices can be added and subtracted. Furthermore, if they have compatible shapes, they can be multiplied. More precisely, given two matrices $A$ and $B$, the matrix $AB$ is defined when the number of columns of $A$ is equal to the number of rows of $B$. In particular, given a natural number $n$, any two matrices $A$ and $B$ with $n$ columns and $n$ rows can be multiplied in both ways (that is, both $AB$ and $BA$ exist).

For questions specifically concerning matrix equations, use the tag.

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The eigenvalue of $A^TA$

If $\lambda$ is the eigenvalue of matrix $A$,what is the eigenvalue of $A^TA$?I have no clue about it. Can anyone help with that?
89085731
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c0mpatible system $A^TAx=A^Tb$

Let $A\in\mathbb{R}^{n\times n}$ be a singular matrix. Prove that the system $$A^TAx=A^Tb$$ is compatible for any $b\in\mathbb{R}^n$. I want to prove that $A^Tb\in Ran(A^TA)$,i.e. $A^Tb\bot Ker(A^TA)$
89085731
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diferences of spectral decomposition of different types of matrices

For an $n \times n$ square complex matrix let say $A$ with eigenvalues $\lambda_1,\lambda_2,.....,\lambda_n$. $A$ is normal iff $A$ is unitary diagonalizable;that is there exist unitary matrix U such that $UAU^*=\text{diag}(\lambda_1,…
hafsah
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Can I calculate this sum using matrix multiplication?

I would like the following sum for the matrices $A,B$, both $n \times n$: \begin{align} \sum(A_{ij} \cdot B_{jk} \cdot A_{kz} \cdot B_{zf}) & \text{ for all } j,k,z,f \text{ such that } i≠k,j≠z,k≠f \end{align} Is there a way to get at this sum using…
adam levin
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Is there would be any matrices $A$ & $B$ where the relation $AB$-$BA$=$I$ holds?

Here , the actual question is to find any matrices $A$ and $B$ such that $AB-BA=I$ relation holds. but actually, I dont think that we could not find any such matrices. As, the diagonal elements of the matrices $A$ and $B$ would be same. so, after…
ROBINSON
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Compute the condition number of matrix A=$\left(\begin{array}{cc} 1 & -1\\0&-1\end{array}\right)$ in norm $||.||_2$

Compute the condition number of matrix A=$\left(\begin{array}{cc} 1 & -1\\0&-1\end{array}\right)$ in norm $||.||_2$. Since $A^{-1}$ is same as $A$. How to dela with that?
89085731
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Prove the matrix is positive definite

Prove that the matrix $\left(\begin{array}{cc} 2+\alpha & -1\\-1&2+\alpha\end{array}\right)$ is positive definite in $\mathbb{C}^2$ for any $\alpha>-1$
89085731
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3x3 matrix $B$ with eigenvalues 0, 1, 2. What information can we deduce from this?

I am given 4 choices for what can be deduced from the eigenvalues of matrix $B$, and 3 of them are correct and I have to choose which ones. The options are: a) The rank or matrix $B$ b) The determinant of $B^TB$ c) The eigenvalues of $B^TB$ d) The…
mangopancake
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Is there a name for a matrix which satisfies det($AA^T$) = det($A^TA$)

Suppose, following is true for a square matrix $A$: $|AA^T| = |A^TA| = |A|^2$ Then, is it some special kind of matrix which has a name?
Rajendra Uppal
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Understanding the derivation of ML-estimator (statistics)

I'm trying to understand the derivation of a ML-estimator and more specifically the rewriting of the covariance matrix Sigma. In this rewriting a lemma is used to show that $$ (1) \hspace{1.4 cm}\Omega=\sigma^2_{c} +…
Sunv
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Schur's theorem

Let, $A$ be a Hermitian matrix of order $n$. Let, diag$(A)$ denote the vestors whose coordinates are the diagonal entries of $A$ and $\lambda$($A$) denote the vector whose coordinates are the eigen values of $A$ specified in any order. Show…
Topology
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Finding Eigen value for a 2 x 2 matrix with only variables

$A = \left[\begin{array}[c]{rr} a & b\\ c & d\end{array}\right]$ Normally to find the Eigen values of a matrix I would simply find $det(A-λI)$ and equate it to $0$, which in this case gives me $λ^2-(a+d)λ+ad-bc = 0$ With so many variables, how am I…
mangopancake
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log series of matrix

I need to prove the following inequality $\log(I-A)<={\|A\|}/({1-\|A\|})$ expansion is same as reals. i tried to apply triangular ineqality but i have a confusion if i can apply it to infinite series or not if yes then its easy .then we can apply…
biswpo
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Tricky matrix basis question

I have two possible answers that I came up with for the basis of the column space for the matrix $\begin{bmatrix}2& 4\\ 4 &8\\ 1 & 2\\ 0 &0\end{bmatrix}$ .. Is it a matrix with the bottom row removed and e1 and e2 added $\begin{bmatrix}2& 0 & 0\\ 4…
user2514676
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polynomial with real coeficients for a matrix

let $A$ belongs to $M_{n \times n}$ then we have to show that there exist a polynomial $f(x)$ with real coefficients such that $f(A)=0$..we know that this is true for characteristic polynomial i.e for every matrix satisfies its characteristic…
hafsah
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