Eigenvalues and diagonalisation

35 problems · Eigenvalues, eigenvectors, characteristic polynomials and diagonalisation.

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Ex.35Eigenspaces are subspaces

#eigenspace #subspace #linear map

Ex.133Similar matrices have the same eigenvalues

#similarity #eigenvalues #characteristic polynomial

Eigenvalues and diagonalisation

Ex.148A matrix with a single eigenvalue

#eigenvalues #eigenspace #diagonalization #Jordan block

Eigenvalues and diagonalisation

Ex.164Trigonalization and eigenvalues

#trigonalization #characteristic polynomial #real eigenvalues

Eigenvalues and diagonalisation

Ex.174Functions of a diagonalizable matrix

#functions of matrices #diagonalization #polynomial #well-definedness

Eigenvalues and diagonalisation

Ex.460Eigenvalues and eigenvectors of a 2×2 matrix

#eigenvalues #eigenvectors #characteristic polynomial

Eigenvalues and diagonalisation

High school

Ex.461A 2×2 matrix with distinct eigenvalues

#eigenvalues #eigenvectors #distinct roots

Eigenvalues and diagonalisation

High school

Ex.697Powers of a matrix

#diagonalisation #matrices

Eigenvalues and diagonalisation

Ex.704Powers of a 2×2 matrix with eigenvalues 1 and 2

#diagonalisation #matrices

Eigenvalues and diagonalisation

Ex.933A 2×2 matrix with a double eigenvalue

#eigenvalues #double root #diagonalizability

Eigenvalues and diagonalisation

High school

Ex.934A 3×3 matrix with distinct eigenvalues

#eigenvalues #eigenvectors #3x3 matrices

Eigenvalues and diagonalisation

Ex.935Eigenvalues of a symmetric matrix

#symmetric matrix #eigenvalues #trace and determinant

Eigenvalues and diagonalisation

Ex.936The characteristic polynomial

#characteristic polynomial #eigenvalues #trace #determinant

Eigenvalues and diagonalisation

Ex.937Eigenvalues of a rotation

#rotation #complex eigenvalues #2x2 matrices

Eigenvalues and diagonalisation

Ex.938Similar matrices have the same eigenvalues

#similarity #eigenvalues #characteristic polynomial

Eigenvalues and diagonalisation

Ex.939Powers via diagonalization

#diagonalization #powers #matrix

Eigenvalues and diagonalisation

Ex.940A basis of an eigenspace

#eigenspace #basis #dimension

Eigenvalues and diagonalisation

Ex.941Sum of eigenvalues equals trace

#eigenvalues #trace #characteristic polynomial

Eigenvalues and diagonalisation

Ex.942Trace and determinant of a 3×3 matrix via eigenvalues

#trace #determinant #eigenvalues

Eigenvalues and diagonalisation

Ex.1658Functions of a diagonalizable matrix

#functions of matrices #diagonalization #polynomial

Eigenvalues and diagonalisation

Ex.1659The diagonalizability criterion

#diagonalizability #multiplicity #eigenspaces

Eigenvalues and diagonalisation

Ex.1660An involution is diagonalizable

#involution #diagonalization #eigenvalues

Eigenvalues and diagonalisation

Ex.1661A projection is diagonalizable

#projection #diagonalization #eigenvalues

Eigenvalues and diagonalisation

Ex.1662Diagonalizing a 2×2 matrix

#diagonalization #change of basis #eigenvalues

Eigenvalues and diagonalisation

Ex.1663Diagonalizing a 3×3 matrix

#diagonalization #eigenspaces #change of basis

Eigenvalues and diagonalisation

Ex.1664Geometric versus algebraic multiplicity

#multiplicities #eigenspace #diagonalization

Eigenvalues and diagonalisation

Ex.1665A matrix with a single eigenvalue

#eigenvalues #eigenspace #diagonalization

Eigenvalues and diagonalisation

Ex.2124The Cayley-Hamilton theorem

#Cayley-Hamilton #characteristic polynomial #matrix identity

Eigenvalues and diagonalisation

Ex.2125Eigenvalues of a circulant matrix

#circulant matrix #eigenvalues #roots of unity

Eigenvalues and diagonalisation

Ex.2126Powers of a companion matrix and Fibonacci numbers

#companion matrix #Fibonacci #diagonalization

Eigenvalues and diagonalisation

Ex.2127Diagonalizability depending on a parameter

#diagonalizability #parameter #eigenspaces

Eigenvalues and diagonalisation

Ex.2128Trigonalization and eigenvalues

#trigonalization #characteristic polynomial #real eigenvalues

Eigenvalues and diagonalisation

Ex.2129Diagonalizability and the minimal polynomial

#minimal polynomial #diagonalization #criterion

Eigenvalues and diagonalisation

Ex.2130Spectral decomposition of a symmetric matrix

#spectral decomposition #symmetric matrix #projection

Eigenvalues and diagonalisation

Ex.2131Orthogonal diagonalization of a symmetric matrix

#symmetric matrix #orthogonal diagonalization #spectral theorem

Eigenvalues and diagonalisation