# 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.

35,069 questions
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### Square matrix geometric meanings

What is the geometric meaning of 1) Determinant of square matrix 2) Inverse of square matrix 3) Trace of square matrix
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### Orthogonal matrices with eigenvalues equally spaced over unit circle?

We know that the eigenvalues of orthogonal matrices have norm 1, and thus they are all on the unit circle. However, I wonder if there is a way to construct a orthogonal matrix (with real entries) ...
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### Another proof for Sherman Morrison Formula?

The proof of Sherman Morrison Formula is on wikipedia as well as this question Proof of the Sherman-Morrison Formula. Isn't there a proof which does not uses multiplication of the inverse and the ...
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### Proof of Jordan-Chevally-Decomposition

Let A be a square matrix over $\mathbb{C}$, prove there are matrices $D$ and $N$ such that $A = D + N$ such that $D$ is diagonalizable, $N$ is Nilpotent and $DN = ND$. I can see that any ...
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### Orbits of the conjugation action of $GL_3 (\mathbb{R})$ on the nonsingular symmetric $3\times3$-matrices

Let $S$ be the space of all symmetric $3 \times 3$ matrices of full rank and with real entries. $GL_3 (\mathbb{R})$ acts on this space by conjugation, \begin{align*} g.A = (g^{-1})^T A g^{-1}, \quad ...
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### For a square matrix A over $\mathbb{C}$, Proofs that matrices D and N exist with A=D+N under different conditions

(i) D is Diagonalizable This one i believe to be fairly straightforward, if D is diagonalizable then we can allow $D^t = I$ (where I is the identity) and therefore D id diagonalizable and therefore A=...
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### Should I use pseudo-inverses to prove this?

Let $A=A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ and $z≥0$. Let $A(x)y≫0$ for $x≠x^*$. Then $z≯0$. How to prove this? Should I use pseudo-inverses?
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### Get from a transformation matrix to the resultant span of the solution set

Guess that's a very basic question, but anyway: I have the following transformation matrix: $$\begin{bmatrix}1 & 0 & 0 & -\tfrac{1}{3}\\ 0 & -6 & -3 & -1\end{bmatrix}$$ And I ...
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### Alternatives in Farka's Lemma as boudaries

I am attempting to solve a problem in the field of Economics, and for that purpose I have devised the following lemmas. Lemma 1: Let $A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ ...
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### If $A$ and $Q$ are unitary, then $U = Q^{-1}AQ$ is unitary…

Here is what I have so far... Since $A$ and $Q$ are unitary, by definition, we have that $AA^* = A^*A = I$ and $QQ^* = Q^*Q = I$, in other words, $A^* = A^{-1}$ and $Q^* = Q^{-1}$. We can then define ...
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### Amount of integer multiplications performed with matrices - how much faster?

I am trying to determine how much faster a multiprocessing system would be at multiplying matrices than a single processor system. Here is my thought process/example: Assume Matrix A is k x l and ...
Let some matrix be $A$, what methods are there for attaining a recurrence relation which allows us to calculate - for example - the leading term in the matrix $A^n$ when given the leading terms of a ...
Consider the matrix: $\begin{bmatrix} 1&0&1&0&0\\0&1&1&1&1\\1&1&1&1&1\\1&1&0&0&0\\1&0&1&0&0\end{bmatrix}$. I checked ...