# Questions tagged [diagonalization]

For questions about matrix diagonalization, that is, writing a matrix, a bilinear form or an operator into a "basis" making this one diagonal. This tag is **NOT** for diagonalization arguments from logic and set theory.

312 questions
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### Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ...
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### When a matrix has same eigenvalues of its column-swapped version?

What are the properties needed for a matrix $A$ to have $\mbox{Spec}(A)= \mbox{Spec}(A \cdot P)$, where P = \begin{pmatrix} 0 & \cdots & 0 & 1 \\ \vdots & \...
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### Why is it useful to know when a linear operator on a vector space is diagonalizable?

I'm currently taking a conceptual course in linear algebra, and I'm trying to understand why it would be theoretically useful or illuminating to know when a linear operator (or its matrix ...
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### Understanding part of the proof of Spectral Theorem for symmetric matrices

I'm reading a textbook where the Spectral Theorem for symmetric matrices is proven. I understand almost everything about the proof except for one thing. The theorem is stated as follows: Theorem: Let ...
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### What can we say about the eigenvalues and diagonalization of this $2N\times2N$ matrix $A$?

There is a $2N\times2N$ matrix $A$, where $N$ is a positive integer, which is of the form: $A=\left(\begin{array}{cc} B & C\\ -C^{*} & -B^{*} \end{array}\right),$ where $B$ is a hermitian ...
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### Diagonalization of matrices of the form BAB

If $A$ and $B$ are real symmetric matrices, the matrix $BAB$ is also a real symmetric matrix. My question is : knowing the orthogonal diagonalization of $A$ and $B$, can we obtain the orthonormal ...
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### Diagonalizability condition involving direct sum of nullspace and range

I'm working on the following problem from Axler's Linear Algebra Done Right (Exercise 5.C.5): Suppose $V$ is a finite-dimensional complex vector space and $T\in \cal{L}(V)$. Prove that $T$ is ...
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### Matrix functions which are not scalar coefficient power series expansions.

Inspired by this question, I started to wondering (firstly) if there exist any (useful / famous) matrix functions which are not defined as power series expansions with scalar coefficients, but rather ...
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### Simultaneous orthogonal diagonalization

Statement Let $A, B \in M_{n\times{}n}(\mathbb{R})$ be symmetric and commutative such that $AB=BA$. Then $A$ and $B$ are simultaneously orthonormally diagonalizable. i.e. there is $Q \in O(n)$ such ...
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### Is a principal submatrix of a diagonalizable matrix diagonalizable?

Let $A$ be diagonalizable, i.e., $A=X \Lambda X^{-1}$ for some diagonal matrix $\Lambda$. Consider $B$ which is a principal submatrix of $A$. Does there exist an invertible matrix $Y$ and a diagonal ...
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### Measure of diagonal dominance

Given any (singular or non-singular) matrix, is there a standard measure for how close the matrix is close to diagonal dominance? For example this matrix: \begin{pmatrix} 5 &2 &2 \\ 5 &1 &...
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### Proof of diagonalizing

Let $k<n$ and $v_{1},v_{2},...,v_{k} \in \mathbb{R}^{n}$ be non-zero vectors, orthogonal with respect to the standard inner product in $\mathbb{R}^{n}$, where the vectors of $\mathbb{R}^{n}$ ...
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### Interpretation of a projection on one eigenspace (from symmetric matrices) to another

I have a reference $N \times N$ symmetric matrix -- with distinct eigenvalues -- decomposed using SVD as: $$R_{ref} = V_{ref} D_{ref} V^{-1}_{ref}$$ If i get a matrix $S$ -- with distinct ...
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### If $A\in\Bbb \{\pm1,0\}^{n\times n}$ is symmetric of rank $<n$, does $A-I$ have rank $n$?

Supposing we have a symmetric matrix $A\in\Bbb \{\pm1,0\}^{n\times n}$ of rank $m<n$ with all $+1$ or all $-1$ or all $0$ as diagonals and no $0$ on non-diagonals, when do we have $A-I$ to be full ...
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### Simultaneous diagonalization of commuting matrix

I have 3 diagonalizable matrices $A,B,C$. They commute with each other $[A,B]=[B,C]=[A,C]=0$  The matrix $A$ is Hermitian but $B$ and $C$ have no properties. [/edit] I can get the eigenvalues ...
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### Smooth family of operators induces a smooth family of projections.

Let $H$ be a Hilbert space, and let $A:U \rightarrow L(H)$ be a smooth$^{1}$ family of unbounded closed operators with common dense domain. Suppose furthermore that $A_{t}$ is normal for each $t \in U$...
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### Show the form of $J$ and $P$ for Leslie matrix $A$ when $A = PJP^{-1}$

I'm trying to solve this for a homework assignment. The Jordan Normal form theorem states that every complex $n \times n$ matrix $A$ van be written as $A=PJP^{-1}$, where $J$ is the diagonal matrix ...
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### How to calc action of operator on functions without diagonalizing the operator?

I have an hermitian operator $$h = -\frac{1}{2} \frac{\partial^2}{\partial x^2} + v(x)$$ acting on square integrable functions that live in some subspace $V$ of $R$. If I want a computer to ...
The definition of probability simplex in $\mathbb{S}^n$: $$\{X\in \mathbb{S}^n: X\geq 0, \text{tr}(X) = 1\},$$ where $\mathbb{S}^n$ is the vector space of symmetric $n\times n$ matrices. A ...