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Eigenvalues and eigenvectors

The characteristic polynomial and the directions a matrix only stretches.

An eigenvector is a direction a matrix does not rotate, only stretches by its eigenvalue. Find eigenvalues as roots of det(A − λI), then eigenvectors by solving (A − λI)v = 0. In the 3D view the eigenvectors are drawn on the transformed cube.

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

Linear-Algebra · Eigenvalues

Eigenvalues of [[2,1],[1,2]]

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

Àwọn ìṣàmúlò-ètò

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

Fi àwọn àgbèwọlé hàn

\lambda = 3,\; \lambda = 1

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Bii o ṣe le: Eigenvalues and eigenvectors

Eigenvalues are the roots of the characteristic polynomial.
Subtract λ from the diagonal.
Expand the determinant.
Factor.
Eigenvalues (with multiplicity).
Solve (A − 1I)v = 0 for a basis eigenvector.
Solve (A − 3I)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 Matrix multiplication Row reduction Vector spaces, span and linear independence Orthogonality, projections and least squares Diagonalisation and matrix powers Linear transformations and change of basis