Diagonalizing a matrix: A = XΛX⁻¹
Slide 2 found that A = 4123 stretches 11 by 5 and 1−2 by 2. Now use those two arrows as your units of measure. Think of a city whose streets run at a slant: directions get easy once you say "3 blocks along this street, 2 blocks along that one". Describe any vector as so many steps of x₁ plus so many steps of x₂. Then A is simple: it multiplies the first count by 5 and the second count by 2, and nothing mixes. Writing this down as matrices is called diagonalizing: the eigenvectors go in the columns of X, the eigenvalues go on the diagonal of Λ, and A = XΛX⁻¹.
Lesson 7 · slide 3A = XΛX⁻¹- X has the eigenvectors as columns; Λ (capital lambda) is diagonal with the eigenvalues, in the same order.
- AX = XΛ says Axᵢ = λᵢxᵢ for every column at once.
- If A is n by n and has n independent eigenvectors, X is invertible and A = XΛX⁻¹.
- Powers become easy: Aᵏ = XΛᵏX⁻¹, and Λᵏ just raises each diagonal entry to the power k.
Counting in eigenvector steps
Take x = 30. It is 2 steps of x₁ = 11 plus 1 step of x₂ = 1−2, because 2·11 + 1·1−2 = 30. Now A acts on each part by itself:
The counts 2 and 1 became 10 and 2: each was multiplied by its own λ. That is all A does, once you measure in eigenvector steps.
All eigenvectors at once: AX = XΛ
Put x₁ and x₂ side by side as the columns of X. Column by column, AX holds Ax₁ = 5x₁ and Ax₂ = 2x₂. That is the same as X times the diagonal matrix Λ, which multiplies column 1 by 5 and column 2 by 2:
Building X and Λ
The recipe has one rule about order: the eigenvalue in position i of Λ goes with column i of X. Swap one, and you must swap the other.
A = XΛX⁻¹, checked
Different eigenvalues always give independent eigenvectors, so X has an inverse. Multiply AX = XΛ on the right by X⁻¹ to get A = XΛX⁻¹. With the 2 by 2 inverse from lesson 3 (det X = −2 − 1 = −3):
Read XΛX⁻¹x from right to left, and it is the street-counting idea. X⁻¹ finds the counts, Λ stretches each count, and X turns the counts back into a vector:
Powers for free
A² = XΛX⁻¹XΛX⁻¹ = XΛ²X⁻¹, since X⁻¹X = I in the middle. Likewise Aᵏ = XΛᵏX⁻¹, and Λᵏ just raises each diagonal entry to the power k:
A¹⁰⁰ is no harder: Λ¹⁰⁰ has 5¹⁰⁰ and 2¹⁰⁰ on its diagonal.
Keep reading with the full course
Finish lesson 7 and you will be able to: tell whether a system settles down or blows up.
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Examples
Example 1 · A diagonal matrix is already done
A = 3007: X = I and Λ = A. The axes are the eigenvectors.
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