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Diagonalisation and matrix powers

A = PDP⁻¹, computing Aⁿ, and the matrices that cannot be diagonalised.

With a full set of eigenvectors, A = PDP⁻¹ where D is diagonal, so Aⁿ = PDⁿP⁻¹: Fibonacci in closed form, Markov chains at equilibrium. Picture it: the transformation stretching along eigenvector directions and nothing else. Think it: symmetric matrices always diagonalise with orthogonal eigenvectors (the spectral theorem); defective ones need Jordan form.

Àwọn ààtò àwọn ìṣàmúlò-ètò · eigenvalues of [[4,1],[2,3]]

Linear-Algebra · Eigenvalues

Eigenvalues of [[4,1],[2,3]]

\left[\begin{matrix}4 & 1\\2 & 3\end{matrix}\right]

Àwọn ìṣàmúlò-ètò

\det(A - \lambda I) = 0
Eigenvalues are the roots of the characteristic polynomial.
\det\left[\begin{matrix}4 - \lambda & 1\\2 & 3 - \lambda\end{matrix}\right] = 0
Subtract λ from the diagonal.
\lambda^{2} - 7 \lambda + 10 = 0
Expand the determinant.
\left(\lambda - 5\right) \left(\lambda - 2\right) = 0
Factor.
\lambda = 5, \lambda = 2
Eigenvalues (with multiplicity).
\lambda = 2:\ (A - \lambda I)\mathbf{v} = 0 \Rightarrow \mathbf{v} = \left[\begin{matrix}- \frac{1}{2}\\1\end{matrix}\right]
Solve (A − 2I)v = 0 for a basis eigenvector.
\lambda = 5:\ (A - \lambda I)\mathbf{v} = 0 \Rightarrow \mathbf{v} = \left[\begin{matrix}1\\1\end{matrix}\right]
Solve (A − 5I)v = 0 for a basis eigenvector.

Fi àwọn àgbèwọlé hàn

\lambda = 5,\; \lambda = 2

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Bii o ṣe le: Diagonalisation and matrix powers

Eigenvalues are the roots of the characteristic polynomial.
Subtract λ from the diagonal.
Expand the determinant.
Factor.
Eigenvalues (with multiplicity).
Solve (A − 2I)v = 0 for a basis eigenvector.
Solve (A − 5I)v = 0 for a basis eigenvector.

Àwọn Àtòjọ-ẹ̀yàn

What does a determinant mean geometrically?

It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.

What is an eigenvector?

A direction the matrix does not turn: it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.

Why is matrix multiplication not commutative?

Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.

Diẹ̀ nínú Linear Algebra

Determinants Matrix inverse Eigenvalues and eigenvectors Matrix multiplication Row reduction Vector spaces, span and linear independence Orthogonality, projections and least squares Linear transformations and change of basis