Lesson 10: Eigenvalues & Eigenvectors

Discussion prompt

Before we open Lesson 10: Eigenvalues & Eigenvectors: without looking back, what was the main idea of The Chain Rule & Backpropagation, and what could you do by the end of it that you could not do before?

Hint: One sentence for the idea, one for the skill. If the second one is blank, that is the part to revisit.

Discussion prompt

One 2×2 matrix carries the first half of this lesson. It is symmetric (A = Aᵀ), which will matter enormously.

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Discussion prompt

Explain The eigen-equation Av = λv by analogy to something with no Machine Learning in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Discussion prompt

Explain Directions of pure stretch to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Predict first

Subtract λ from each diagonal entry of A (that is what −λI does) and write the matrix out:

Commit before you compute: what does Form A − λI, then its determinant come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Determinant of a 2×2 is (top-left)(bottom-right) − (top-right)(bottom-left)

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. The standard det[[a,b],[c,d]] = ad − bc rule.

Annotate

From Form A − λI, then its determinant — read this one piece at a time. What is each part doing?

On: \( A - \lambda I = \begin{bmatrix} 2 - \lambda & 1 \\ 1 & 2 - \lambda \end{bmatrix} \)

• λI has λ on the diagonal and 0 elsewhere, so only the diagonal entries change.
• The standard det[[a,b],[c,d]] = ad − bc rule.

Put in order

Put the moves of Expand to the characteristic polynomial into the order they have to happen.

1. Multiply out (2 − λ)²
2. Collect constants: 4 − 1 = 3
3. Factor: two numbers multiplying to 3, adding to 4

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. (2 − λ)² = 4 − 4λ + λ². FOIL, no shortcut.

Draw it

Draw what Expand to the characteristic polynomial just did — the shape of it, not the line-by-line working. One picture, labels only where you need them. Then check it against the steps: anything you could not draw is a step you followed rather than understood.

Discussion prompt

Eigenvector for λ = 3 — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Substitute λ = 3 into A − λI

Discussion prompt

Work backwards. The example finished here:

Pick v = [1, 1], then normalize to unit length

What was it asked to do, and what must it have been given? Reconstruct the problem from its answer.

Hint: Every quantity in the result had to enter somewhere. Account for each one.

Put in order

Put the moves of Eigenvector for λ = 1 into the order they have to happen.

1. Substitute λ = 1: diagonal becomes 2 − 1 = 1
2. Row 1 says v₁ + v₂ = 0, so v₁ = −v₂
3. Pick v = [1, −1], normalize by √2

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. The two components are opposite.

Draw it

Translate both ways. First write the expression above as a sentence with no symbols in it at all. Then cover it, and write your sentence back as notation. If the two versions disagree, the disagreement is the thing to fix.

Discussion prompt

Never trust an eigenpair you haven't plugged back in. Multiply A by each v and confirm you get λv. This snippet is complete and runnable:

What do you need to know — or decide — before the first line can be written? List everything the problem has to hand you.

Hint: Anything you would have to invent to get started is a thing the problem must supply.

Sort into buckets

Sort these by value (verified), from memory, without looking back at Verify Av = λv, and check against NumPy. Telling them apart on the spot is the skill; the table is only where the answer happens to be written down.

[2.121, 2.121]

A @ v3; 3 * v3

[1.0, 3.0]

np.linalg.eigh vals

[[−0.707, 0.707], [0.707, 0.707]]

eigh vecs (columns)

g1

value (verified) is "[2.121, 2.121]" for A @ v3, 3 * v3 — that is what the table on "Verify Av = λv, and check against NumPy" records, and it is the single property separating this group from the rest.

g2

value (verified) is "[1.0, 3.0]" for np.linalg.eigh vals — that is what the table on "Verify Av = λv, and check against NumPy" records, and it is the single property separating this group from the rest.

g3

value (verified) is "[[−0.707, 0.707], [0.707, 0.707]]" for eigh vecs (columns) — that is what the table on "Verify Av = λv, and check against NumPy" records, and it is the single property separating this group from the rest.

Predict first

A student writes this, and it looks reasonable:

The eigenvalues of [[3,1],[0,2]] turn out to be 3 and 2 — its diagonal. So just read eigenvalues off the diagonal of any matrix.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: Generalizes a rule that only holds for TRIANGULAR matrices.

Diagonal = eigenvalues only for triangular (or diagonal) matrices, where the determinant is the product of diagonal entries. Otherwise solve det(A − λI) = 0.

Why: Generalizes a rule that only holds for TRIANGULAR matrices. A is not triangular, and we just proved its eigenvalues are 1 and 3 — nowhere near 2 and 2.

Predict first

The table runs: V @ diag(vals) @ V.T | [[2., 1.], [1., 2.]] · np.allclose(recon, A) | True

In Reconstruct A from its eigen-decomposition, given the rows so far: what is the next one — the row where quantity is V.T @ V?

Correct: V.T @ V | [[1., 0.], [0., 1.]] (identity)

Why: The relationship between the columns, not the individual numbers, is what generates the next row. np.allclose prints True. The decomposition is faithful: A really is 'rotate into eigen-axes (Vᵀ), stretch by Λ, rotate back (V)'.

Comparison matrix

From Reconstruct A from its eigen-decomposition: refill the value (verified) column from what you know. The rest of the table is as it appeared.

Comparison matrix

From Eigenvalues classify a symmetric matrix: every row here is a choice with a cost. Fill the quadratic zᵀAz column, then say which row you would actually pick and what you give up for it.

Fill in the blanks

From A non-symmetric matrix behaves differently — one line has had its right-hand side removed. Put it back.

import numpy as np
B = np.array([[3., 1.], [0., 2.]])
vals, vecs = np.linalg.eig(B) # general solver
print(vals) # [3., 2.]
print(vecs)
v_a = vecs[:, 0]; v_b = vecs[:, 1]
print(v_a @ v_b) # dot product != 0

Why: v_a is what everything below it consumes, so the wrong expression here fails later and somewhere else. Non-symmetric ⇒ the spectral theorem does NOT apply ⇒ no orthogonality guarantee.

Predict first

A student writes this, and it looks reasonable:

Eigenvectors form the natural axes of a transform, so they must be perpendicular — orthogonalize or dot them freely.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: We just measured it fail: B = [[3,1],[0,2]] has eigenvectors [1,0] and [−0.707,0.707] with dot product 0.707.

Orthonormal eigenvectors are guaranteed only by the spectral theorem — i.e. for symmetric A.

Why: We just measured it fail: B = [[3,1],[0,2]] has eigenvectors [1,0] and [−0.707,0.707] with dot product 0.707. Orthogonality is not automatic.

Discussion prompt

The rule this trap just fixed:

Orthonormal eigenvectors are guaranteed only by the spectral theorem — i.e. for symmetric A.

Now break it on purpose. Build a case that violates it and follow the consequences until something visibly fails. Where does the failure first show up — and would you have noticed it if you had not been looking?

Hint: The dangerous rules are the ones whose violation still produces an answer. If yours fails loudly, try to find one that fails quietly.

Predict first

Write the start b₀ as a blend of eigenvectors: b₀ = c₃v₃ + c₁v₁ (dominant v₃ with λ=3, other v₁ with λ=1). Applying A scales each piece by its own eigenvalue:

Commit before you compute: what does Why it converges: work in the eigenbasis come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: For b₀ = [1,0]: ratio small/big = 1/3, 1/9, 1/27, …

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. Verified: 0.333, 0.111, 0.037, 0.0041 at k = 1,2,3,5.

Step through it

Step through Why it converges: work in the eigenbasis one row at a time. One of these columns never changes — find it, and say why it cannot.

1. Step 1: k is 1
2. Step 2: k is 2
3. Step 3: k is 3
4. Step 4: k is 5

Fill in the blanks

From Trace power iteration on A, step by step — one line has had its right-hand side removed. Put it back.

import numpy as np
A = np.array([[2., 1.], [1., 2.]])
b = np.array([1., 0.])
for i in range(6):
Ab = A @ b
b = Ab / np.linalg.norm(Ab)
print(i + 1, b.round(4), round(b @ A @ b, 5))

Why: Ab is what everything below it consumes, so the wrong expression here fails later and somewhere else. Divide [2,1] by 2.236. Rayleigh bᵀAb = 2.8, already climbing toward 3.

Annotate

Walk the callouts on Trace power iteration on A, step by step. Each one is a place this is easy to get subtly wrong.

• Divide [2,1] by 2.236. Rayleigh bᵀAb = 2.8, already climbing toward 3.
• Exactly the dominant eigenpair we solved by hand. The Rayleigh quotient converges even faster than the vector — it's accurate to 5 decimals by step 6.

Fill in the blanks

From Deflate A and recover the second eigenpair — one line has had its right-hand side removed. Put it back.

import numpy as np
A = np.array([[2., 1.], [1., 2.]])
v3 = np.array([1., 1.]) / np.sqrt(2)
A_def = A - 3.0 * np.outer(v3, v3) # remove the lambda=3 piece
b = np.array([1., 0.])
for _ in range(30):
b = A_def @ b
b = b / np.linalg.norm(b)
print(A_def)
print(b.round(4), round(b @ A_def @ b, 4))

Why: b is what everything below it consumes, so the wrong expression here fails later and somewhere else. The λ=3 direction is now dead (eigenvalue 0), so the λ=1 direction dominates A'.

Predict first

A student writes this, and it looks reasonable:

Power iteration is just 'multiply by A a lot', so loop b = A @ b and read off the direction at the end — skip the divide.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: Each multiply scales the dominant part by ~3, so after 30 steps entries are ~3³⁰ ≈ 2e14 and keep growing — eventually overflow to inf, and the eigenvalue read-out bᵀAb explodes instead of settling.

Divide by the length every iteration so the vector stays unit-sized — we only care about its direction.

Why: Each multiply scales the dominant part by ~3, so after 30 steps entries are ~3³⁰ ≈ 2e14 and keep growing — eventually overflow to inf, and the eigenvalue read-out bᵀAb explodes instead of settling.

Step through it

Step through Our running dataset one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: point is 1
2. Step 2: point is 2
3. Step 3: point is 3
4. Step 4: point is …
5. Step 5: point is 10

Discussion prompt

Explain The covariance matrix to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Discussion prompt

Read the picture before the words. What is this showing, and what is the one thing it is built to make obvious? Commit to an answer, then read on.

Hint: Name the parts, then say what changes between them — and if nothing changes, say what is being held still.

Discussion prompt

Explain What the covariance eigenvectors point at by analogy to something with no Machine Learning in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Discussion prompt

Build the covariance entry by entry — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Top-left = Σ(xᶜ)² / 9 = 5.549 / 9 = 0.6166

Draw it

Draw what Build the covariance entry by entry just did — the shape of it, not the line-by-line working. One picture, labels only where you need them. Then check it against the steps: anything you could not draw is a step you followed rather than understood.

Put in order

Put the moves of Eigenvalues of C by the characteristic polynomial into the order they have to happen.

1. Characteristic polynomial: λ² − tr·λ + det = 0
2. Quadratic formula, discriminant = 1.3331² − 4·0.06302 = 1.5251
3. λ₁ = 1.2840 (PC1 variance), λ₂ = 0.0491 (PC2 variance)

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. For a 2×2, det(C − λI) = λ² − tr(C)λ + det(C).

Annotate

From Eigenvalues of C by the characteristic polynomial — read this one piece at a time. What is each part doing?

On: \( \lambda = \frac{1.3331 \pm 1.2349}{2} = 1.2840 \;\text{or}\; 0.0491 \)

• For a 2×2, det(C − λI) = λ² − tr(C)λ + det(C). We derived this exact shape in Part 1.
• Matches np.linalg.eigh exactly. PC1 holds 1.2840/(1.2840+0.0491) = 96.3% of the total variance.

Predict first

The table runs: cov | [[0.6166, 0.6154], [0.6154, 0.7166]] · eigenvalues (desc) | [1.2840, 0.0491] · PC1 direction | [0.6779, 0.7352]

In PCA from scratch, top to bottom, given the rows so far: what is the next one — the row where quantity is PC1 share of variance?

Correct: PC1 share of variance | 1.2840 / 1.3331 = 96.3%

Why: The relationship between the columns, not the individual numbers, is what generates the next row. eigh returns ascending, so we reverse the order to put the biggest-variance direction first.

Discussion prompt

sklearn's PCA centers internally and reports explained_variance_ (the eigenvalues) and components_ (the eigenvectors as rows). Confirm they agree with our from-scratch numbers:

What do you need to know — or decide — before the first line can be written? List everything the problem has to hand you.

Hint: Anything you would have to invent to get started is a thing the problem must supply.

Comparison matrix

From Match it against sklearn: refill the sklearn PCA column from what you know. The rest of the table is as it appeared.

Predict first

The eigenvalue is a promise: project the centered data onto PC1 and the projected scores should have variance equal to λ₁ = 1.2840. Check it:

Commit before you compute: what does Projecting onto PC1 recovers the variance come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Variance of the PC1 scores = 1.2840 = λ₁

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. The scores spread out with exactly the eigenvalue's worth of variance — the concrete meaning of 'eigenvalue = variance along that component.'

Discussion prompt

Work backwards. The example finished here:

Variance of the PC1 scores = 1.2840 = λ₁

What was it asked to do, and what must it have been given? Reconstruct the problem from its answer.

Hint: Every quantity in the result had to enter somewhere. Account for each one.

Predict first

A student writes this, and it looks reasonable:

Just eigen-decompose XᵀX/(n−1) on the raw data — skip the mean subtraction, it's an extra step.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: That top eigenvector comes out [0.686, 0.727] — which is essentially the data's MEAN direction [0.688, 0.726], not its direction of spread.

Variance is defined around the mean, so center first — always.

Why: That top eigenvector comes out [0.686, 0.727] — which is essentially the data's MEAN direction [0.688, 0.726], not its direction of spread. The 'variance' 8.977 is inflated garbage; the real PC1 variance is 1.284.

Sort into buckets

Some of these hold up and some are the exact mistakes this lesson is built to prevent. Sort them.

Holds up

One 2×2 matrix carries the first half of this lesson. It is symmetric (A = Aᵀ), which will matter enormously.; An eigenvector v of A is a nonzero vector that A only scales — it keeps its direction. The scale factor λ is the eigenvalue for that vector.; λ > 1 stretches, 0 < λ < 1 compresses, λ < 0 flips the arrow around, and λ = 0 collapses that direction to a point — the mark of a singular matrix.

Breaks

The eigenvalues of [[3,1],[0,2]] turn out to be 3 and 2 — its diagonal. So just read eigenvalues off the diagonal of any matrix.; Eigenvectors form the natural axes of a transform, so they must be perpendicular — orthogonalize or dot them freely.

sound

These are stated as this lesson states them — each one survives the edge cases Lesson 10: Eigenvalues & Eigenvectors puts it through.

flawed

Each of these is lifted from a trap in this deck: reasonable-sounding, and wrong in a way that only shows up once you rely on it.

Discussion prompt

Run The eigen-toolkit with this step confiscated:

Symmetric A: use np.linalg.eigh → real eigenvalues, orthonormal vectors, A = VΛVᵀ (verify by reconstruction)

Is it still possible? If it is, say what takes its place and what it costs you. If it is not, say exactly what that step was providing that nothing else does.

Hint: A step you can drop for free was never load-bearing. If you cannot drop it, name the thing that goes wrong the moment it is gone.

Discussion prompt

The eigen-toolkit works on the cases you have just seen. Push it to the edge: what is the most degenerate input it still handles — empty, zero, one item, everything equal — and what is the first case where it stops being true? Name the case, not just "it breaks".

Hint: Try the smallest legal input, then the largest, then the one where two things collide. Methods are specified at their edges; the middle takes care of itself.

Eliminate the wrong options

The eigenvalues of A = [[4,2],[1,3]] are the roots of which equation?

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. (4−λ)(3−λ) − 2·1 = 0
• B. (4−λ)(3−λ) = 0
• C. (4−λ) + (3−λ) = 0
• D. 4·3 − 2·1 − λ = 0

Survives elimination: A

Why: det(A − λI) = (4−λ)(3−λ) − (2)(1) = λ² − 7λ + 10 = 0, giving λ = 2 and 5. The off-diagonal product 2·1 must be subtracted, and the trace 7 / det 10 checks confirm it.

Check your understanding

The eigenvalues of A = [[4,2],[1,3]] are the roots of which equation?

• A. (4−λ)(3−λ) − 2·1 = 0 (correct)
• B. (4−λ)(3−λ) = 0
• C. (4−λ) + (3−λ) = 0
• D. 4·3 − 2·1 − λ = 0

Answer: A

Why: det(A − λI) = (4−λ)(3−λ) − (2)(1) = λ² − 7λ + 10 = 0, giving λ = 2 and 5. The off-diagonal product 2·1 must be subtracted, and the trace 7 / det 10 checks confirm it.

Why B tempts people

Forgets the off-diagonal term. (4−λ)(3−λ) = 0 would be correct only if the off-diagonal entries were zero — i.e. a triangular matrix. Here they aren't.

Why C tempts people

That's the trace condition (λ₁ + λ₂ = 7), a single check on the eigenvalues, not the equation that produces them. It gives one linear equation, not the required quadratic.

Why D tempts people

Confuses determinant with the characteristic polynomial. det(A) = 10 is the PRODUCT of the eigenvalues, a check — not an equation you solve for λ this way.

Predict first

Which property is guaranteed for a real SYMMETRIC matrix but NOT for a general square matrix?

Answer it in your own words, now, with nothing to choose from. The options are on the next slide — and picking the right one off a list is an easier skill than producing it.

Correct: Real eigenvalues and orthonormal eigenvectors

Why: The spectral theorem guarantees real eigenvalues and a full orthonormal eigenbasis for symmetric matrices. A general matrix can have complex eigenvalues and non-orthogonal (or too few) eigenvectors.

Check your understanding

Which property is guaranteed for a real SYMMETRIC matrix but NOT for a general square matrix?

• A. Real eigenvalues and orthonormal eigenvectors (correct)
• B. All eigenvalues are strictly positive
• C. The eigenvalues equal the diagonal entries
• D. It has exactly one eigenvector

Answer: A

Why: The spectral theorem guarantees real eigenvalues and a full orthonormal eigenbasis for symmetric matrices. A general matrix can have complex eigenvalues and non-orthogonal (or too few) eigenvectors.

Why B tempts people

Positive eigenvalues is positive-DEFINITENESS, a strictly stronger condition. Symmetric matrices can have negative eigenvalues — e.g. [[0,1],[1,0]] is symmetric with eigenvalues −1 and 1.

Why C tempts people

Diagonal = eigenvalues holds for TRIANGULAR matrices, not symmetric ones. Our symmetric [[2,1],[1,2]] has eigenvalues 1 and 3, not 2 and 2.

Why D tempts people

The opposite is true: a symmetric n×n matrix has a FULL set of n orthonormal eigenvectors, not just one.

Predict first

Power iteration bₖ₊₁ = Abₖ/‖Abₖ‖ converges to the eigenvector with…

Answer it in your own words, now, with nothing to choose from. The options are on the next slide — and picking the right one off a list is an easier skill than producing it.

Correct: the largest eigenvalue in absolute value (the dominant one)

Why: Writing b₀ in the eigenbasis, each multiply by A scales the component along λᵢ by λᵢ. The largest-|λ| component grows fastest and dominates, so b converges to that dominant eigenvector; the eigenvalue follows from bᵀAb.

Check your understanding

Power iteration bₖ₊₁ = Abₖ/‖Abₖ‖ converges to the eigenvector with…

• A. the largest eigenvalue in absolute value (the dominant one) (correct)
• B. the smallest eigenvalue
• C. the eigenvalue closest to zero
• D. whichever eigenvector you started nearest to

Answer: A

Why: Writing b₀ in the eigenbasis, each multiply by A scales the component along λᵢ by λᵢ. The largest-|λ| component grows fastest and dominates, so b converges to that dominant eigenvector; the eigenvalue follows from bᵀAb.

Why B tempts people

The smallest eigenvalue's component shrinks relative to the others every step, so it's the first to vanish — not what power iteration lands on. (To target the smallest, you'd iterate with A⁻¹ — inverse iteration.)

Why C tempts people

Same misconception as B: the near-zero-eigenvalue direction decays fastest under repeated multiplication by A, so iteration moves away from it.

Why D tempts people

The start only affects the coefficients c_i (and how many steps you need). As long as the dominant component isn't exactly zero, iteration converges to the dominant eigenvector regardless of where you began.

Eliminate the wrong options

In PCA, the eigenvalues of the (centered) covariance matrix represent…

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. The variance captured along each principal component
• B. The mean of the data along each axis
• C. The number of samples in each cluster
• D. The reconstruction error of the model

Survives elimination: A

Why: Each eigenvalue is the variance of the data projected onto its eigenvector (principal component) — we verified it: projecting onto PC1 gave scores with variance 1.2840, exactly λ₁. The largest eigenvalue marks the direction of maximum spread.

Check your understanding

In PCA, the eigenvalues of the (centered) covariance matrix represent…

• A. The variance captured along each principal component (correct)
• B. The mean of the data along each axis
• C. The number of samples in each cluster
• D. The reconstruction error of the model

Answer: A

Why: Each eigenvalue is the variance of the data projected onto its eigenvector (principal component) — we verified it: projecting onto PC1 gave scores with variance 1.2840, exactly λ₁. The largest eigenvalue marks the direction of maximum spread.

Why B tempts people

The mean is removed by centering BEFORE the covariance is formed, so it can't be an eigenvalue. Eigenvalues describe spread, not location. (Skipping centering is the exact trap that made the top 'eigenvalue' point at the mean.)

Why C tempts people

PCA is not clustering and knows nothing about cluster membership; eigenvalues are variances (real numbers), not integer counts.

Why D tempts people

Reconstruction error is the sum of the DISCARDED eigenvalues (variance you throw away), not the eigenvalues you keep.

Discussion prompt

Build the dominant-eigenvector finder, then PCA from scratch, and prove both against NumPy/sklearn. You've derived every piece — now assemble them yourself.

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Comparison matrix

From Milestone 4 — deflate for PC2: every row here is a choice with a cost. Fill the value column, then say which row you would actually pick and what you give up for it.

Comparison matrix

From The full program: refill the value (verified) column from what you know. The rest of the table is as it appeared.

Draw it

One page, no notation unless you need it: draw how these connect — The eigen-equation · The spectral theorem · Power iteration · Eigenvectors are PCA · The eigen-toolkit · Your turn: build it. Put an arrow wherever one of them is what makes another possible, and label the arrow with why.