Determine the Singular Values of a Matrix

1. Formula
   Observation

   The slide states "Let A be an m x n matrix" and later gives the example matrix $A = [[1, -1, 3], [3, 1, 1]]$.

A

A general m x n matrix; in the worked example, the specific 2 x 3 matrix [[1, -1, 3], [3, 1, 1]].

m x n matrices in general; 2 x 3 matrices in the example

1. Formula
   Observation

   The opening statement reads "Let A be an m x n matrix."

m, n

Dimensions of the matrix A, with m rows and n columns.

Positive integers

1. Formula
   Observation

   The slide uses $A^T$ in the expressions $A^T A$ and AA^T.

2. Audio
   Observation

   The narrator says "A transpose times A or A times A transpose."

$A^T$

The transpose of A, obtained by turning rows into columns.

n x m matrix when A is m x n

1. Formula
   Observation

   The definition line says singular values are the square roots of the positive eigenvalues of $A^T A$ or AA^T.

positive eigenvalues of $A^T A$ or AA^T

Eigenvalues used to compute singular values; only the positive ones are taken for square roots.

Spectrum of the symmetric Gram matrices $A^T A$ and AA^T

1. Formula
   Observation

   The recall line defines an eigenvalue as a scalar lambda satisfying det(A - lambda I) = 0 or det(lambda I - A) = 0.

2. Audio
   Observation

   The narrator says "an eigenvalue ... is a scalar lambda such that the determinant ... equals zero."

lambda

Scalar eigenvalue variable in the characteristic equation.

Scalars over the underlying field, here real numbers

1. Formula
   Observation

   The formula uses lambda I inside det(A - lambda I).

I

Identity matrix of the same size as the matrix whose eigenvalues are being computed.

Square identity matrix

1. Formula
   Observation

   The characteristic equations are written using det(A - lambda I) and det(lambda I - A).

det

Determinant operator applied to a square matrix.

Square matrices

1. Formula
   Observation

   The worked example computes AA^$T = [[11, 5], [5, 11]]$.

AA^T

In the example, the 2 x 2 product matrix [[11, 5], [5, 11]] whose eigenvalues are found next.

2 x 2 real symmetric matrix in this example

1. Formula
   Observation

   The final displayed line on the slide is lambda = 16 or lambda = 6.

2. Audio
   Observation

   The narrator concludes "resulting in two eigenvalues where we have lambda equals 16" before the clip ends.

1. The spoken sentence reaches only lambda = 16 before the clip ends, but the displayed line also shows lambda = 6.

lambda = 16 or lambda = 6

The two eigenvalues obtained from the characteristic polynomial of AA^T in the example.

Real scalars

1. Formula
   Observation

   The matrix A is shown as [[1, -1, 3], [3, 1, 1]].

A

The original 2x3 matrix whose singular values are being determined.

2x3 real matrix

1. Formula
   Observation

   The product AA^T is computed and shown as [[11, 5], [5, 11]].

AA^T

The product of matrix A and its transpose, used to find eigenvalues for singular values.

2x2 real symmetric matrix

1. Formula
   Observation

   The variable lambda is used in the characteristic equation det(A - lambda I) = 0.

$\lambda$

Eigenvalue of the matrix AA^T.

Real scalar

1. Formula
   Observation

   Slide text: "The singular values of A are the square roots of the positive eigenvalues of $A^T A$ or AA^T."

2. Audio
   Observation

   Narrator repeats the same definition verbatim.

For an m x n matrix A, the singular values are defined as the square roots of the positive eigenvalues of either $A^T A$ or AA^T. The video presents these two Gram matrices as interchangeable sources for the same positive eigenvalues.

1. A is an m x n matrix.

2. Only positive eigenvalues are used in the square-root step.

1. Computing the transpose of a matrix

1. Formula
   Observation

   Slide text: "$A^T A$ and AA^T have the same positive eigenvalues."

2. Audio
   Observation

   Narrator says they "will always have the same positive eigenvalues."

1. The video states the fact but does not prove it within this clip.

The lesson states that $A^T A$ and AA^T share the same positive eigenvalues, so one may choose whichever product is computationally easier when finding singular values.

1. A is an m x n matrix.

2. The claim is about positive eigenvalues only.

1. Definition of singular values via positive eigenvalues

1. Audio
   Observation

   Narrator explains that the first row of A becomes the first column of $A^T$ and the second row becomes the second column.

2. Formula
   Observation

   Example shows $A = [[1,-1,3],[3,1,1]]$ and $A^T = [[1,3],[-1,1],[3,1]]$.

To form $A^T$, rewrite each row of A as a column. In the example, the row (1, -1, 3) becomes the first column and the row (3, 1, 1) becomes the second column.

1. Applies to any rectangular matrix.

1. Audio
   Observation

   Narrator notes that AA^T is a 2 x 2 matrix and therefore chooses it for the eigenvalue computation.

2. Formula
   Observation

   Slide text: "Let's find the eigenvalues of AA^T since AA^T is a 2 by 2 matrix."

After computing both $A^T A$ and AA^T, the video selects AA^T because it is 2 x 2 rather than 3 x 3. This reduces the size of the characteristic polynomial that must be solved.

1. Useful when one of m,n is smaller than the other.

2. Relies on the stated equality of positive eigenvalues.

1. Equality of positive eigenvalues of $A^T A$ and AA^T

1. Formula
   Observation

   Slide text: "Recall: An eigenvalue of A is a scalar lambda s.t. det(A - lambda I) = 0 or det(lambda I - A) = 0."

2. Audio
   Observation

   Narrator reads the same recall statement.

An eigenvalue of a square matrix A is a scalar lambda for which the determinant of A - lambda I vanishes. The video also gives the equivalent form det(lambda I - A)=0.

1. A must be square.

2. I is the identity matrix of the same size as A.

1. Definition of singular values via positive eigenvalues

1. Formula
   Observation

   The text 'The singular values are A are: sigma_i = sqrt(lambda_i)' is displayed.

2. Audio
   Observation

   Narrator states: 'The singular values of A are sigma sub i equal to the square root of lambda sub i'.

The singular values of a matrix A are defined as the square roots of the positive eigenvalues of the matrix product AA^T (or A^TA). In this context, sigma_i = sqrt(lambda_i), where lambda_i are the eigenvalues of AA^T.

1. lambda_i must be non-negative

2. lambda_i are eigenvalues of AA^T

1. Characteristic Equation for Eigenvalues

1. Formula
   Observation

   Text reads: 'Recall: An eigenvalue of A is a scalar lambda s.t. det(A - lambda I) = 0 or det(lambda I - A) = 0.'

2. Formula
   Observation

   Step-by-step derivation shows det([[11, 5], [5, 11]] - lambda[[1, 0], [0, 1]]) = 0 leading to (11-lambda)(11-lambda) - (5)(5) = 0.

To find the eigenvalues of a square matrix M, one solves the characteristic equation det(M - lambda I) = 0. For the 2x2 matrix AA^T, this expands to a quadratic polynomial in lambda.

1. M is a square matrix

2. I is the identity matrix of same dimension

1. Formula
   Observation

   The calculation [[1, -1, 3], [3, 1, 1]] * [[1, 3], [-1, 1], [3, 1]] = [[11, 5], [5, 11]] is shown.

The matrix product AA^T is calculated by multiplying the rows of A by the columns of $A^T$ (which are the rows of A). This results in a symmetric 2x2 matrix.

1. A is a real matrix

1. Formula
   Observation

   Displayed bullet: "$A^T A$ and AA^T have the same positive eigenvalues."

2. Audio
   Observation

   Narrator states the same fact.

1. No proof is provided in this clip.

For an m x n matrix A, the matrices $A^T A$ and AA^T have the same positive eigenvalues.

1. A is an m x n matrix.

For all matrices A of size m x n, the positive parts of the spectra of $A^T A$ and AA^T coincide.

1. Formula
   Observation

   Recall line on slide gives det(A - lambda I)=0 or det(lambda I - A)=0.

2. Audio
   Observation

   Narrator reads the same characterization.

A scalar lambda is an eigenvalue of a square matrix A exactly when det(A - lambda I)=0, equivalently det(lambda I - A)=0.

1. A is square.

2. I is the identity matrix of the same size as A.

For every square matrix A and scalar lambda, lambda is an eigenvalue if and only if the displayed determinant condition holds.

1. Audio
   Observation

   Narrator says: 'Notice both are positive eigenvalues, and therefore we use these values to determine the singular values...'

2. Formula
   Observation

   Screen shows lambda = 16 or lambda = 6.

For the given matrix A, the eigenvalues of AA^T are 16 and 6, which are both positive.

1. $A = [[1, -1, 3], [3, 1, 1]]$

Existential (for this specific A)

1. Caption evidence
   Observation

   A reminder box appears stating: 'Reminder: Only use the positive eigenvalues to find the singular values.'

Only positive eigenvalues of AA^T are used to calculate the singular values of A.

Universal (implied rule)

1. Formula
   Observation

   The slide successively displays $A^T$, then $A^T A = [[10,2,6],[2,2,-2],[6,-2,10]]$, then AA^$T = [[11,5],[5,11]]$.

2. Audio
   Observation

   Narrator explains the transpose rule and comments on the resulting matrix sizes.

1. Expression

   $$A=\begin{bmatrix}1&-1&3\\3&1&1\end{bmatrix}$$
   Explanation

   Start from the given 2 x 3 example matrix.

   Justification

   Given in the problem statement.

   Shown in the video
2. Expression

   $$A^T=\begin{bmatrix}1&3\\-1&1\\3&1\end{bmatrix}$$
   Explanation

   Convert each row of A into a column of $A^T$.

   Justification

   Definition of matrix transpose.

   Shown in the video
3. Expression

   $$A^TA=\begin{bmatrix}1&3\\-1&1\\3&1\end{bmatrix}\begin{bmatrix}1&-1&3\\3&1&1\end{bmatrix}=\begin{bmatrix}10&2&6\\2&2&-2\\6&-2&10\end{bmatrix}$$
   Explanation

   Multiply $A^T$ by A to obtain the 3 x 3 Gram matrix.

   Justification

   Matrix multiplication.

   Shown in the video
4. Expression

   $$AA^T=\begin{bmatrix}1&-1&3\\3&1&1\end{bmatrix}\begin{bmatrix}1&3\\-1&1\\3&1\end{bmatrix}=\begin{bmatrix}11&5\\5&11\end{bmatrix}$$
   Explanation

   Multiply A by $A^T$ to obtain the 2 x 2 Gram matrix.

   Justification

   Matrix multiplication.

   Shown in the video

For the example, $A^T A$ is 3 x 3 and AA^T is 2 x 2, so the smaller matrix AA^T is selected for the eigenvalue step.

1. Formula
   Observation

   The slide shows det([[11,5],[5,11]] - lambda[[1,0],[0,1]]) = 0, then simplifies to (11-lambda)(11-lambda)-(5)(5)=0, then to lambda^2-22lambda+$96=0$, then to (lambda-16)(lambda-6)=0, and finally lambda=16 or lambda=6.

2. Audio
   Observation

   Narrator verbally follows the same algebraic sequence.

1. Expression

   $$\det\left(\begin{bmatrix}11&5\\5&11\end{bmatrix}-\lambda\begin{bmatrix}1&0\\0&1\end{bmatrix}\right)=0$$
   Explanation

   Set up the characteristic equation for the 2 x 2 matrix AA^T.

   Justification

   Eigenvalue definition via det(A - lambda I)=0.

   Shown in the video
2. Expression

   $$\det\left(\begin{bmatrix}11-\lambda&5\\5&11-\lambda\end{bmatrix}\right)=0$$
   Explanation

   Subtract lambda I entrywise from AA^T.

   Justification

   Scalar multiplication and matrix subtraction.

   Shown in the video
3. Expression

   $$(11-\lambda)(11-\lambda)-(5)(5)=0$$
   Explanation

   Evaluate the 2 x 2 determinant.

   Justification

   Determinant formula for a 2 x 2 matrix.

   Shown in the video
4. Expression

   $$121-22\lambda+\lambda^2-25=0$$
   Explanation

   Expand the square and simplify the constant term.

   Justification

   Algebraic expansion.

   Shown in the video
5. Expression

   $$\lambda^2-22\lambda+96=0$$
   Explanation

   Combine like terms to get the quadratic characteristic polynomial.

   Justification

   Arithmetic simplification.

   Shown in the video
6. Expression

   $$(\lambda-16)(\lambda-6)=0$$
   Explanation

   Factor the quadratic.

   Justification

   Factoring a monic quadratic with roots summing to 22 and product 96.

   Shown in the video
7. Expression

   $$\lambda=16\quad\text{or}\quad \lambda=6$$
   Explanation

   Read off the two eigenvalues from the factored equation.

   Justification

   Zero-product property.

   Shown in the video

The eigenvalues of AA^T in the example are 16 and 6.

1. Formula
   Observation

   Visual steps show expansion of determinant: (11-lambda)(11-lambda) - $25 = 0$ -> 121 - 22lambda + lambda^$2 - 25 = 0$ -> lambda^2 - 22lambda + $96 = 0$ -> (lambda-16)(lambda-6) = 0.

1. Expression

   $$\det\left(\begin{bmatrix} 11 & 5 \\ 5 & 11 \end{bmatrix} - \lambda \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\right) = 0$$
   Explanation

   Set up the characteristic equation for the matrix AA^T.

   Justification

   Definition of eigenvalues via determinant.

   Shown in the video
2. Expression

   $$(11 - \lambda)(11 - \lambda) - (5)(5) = 0$$
   Explanation

   Compute the determinant of the resulting 2x2 matrix.

   Justification

   Formula for 2x2 determinant ad-bc.

   Shown in the video
3. Expression

   $$121 - 22\lambda + \lambda^2 - 25 = 0$$
   Explanation

   Expand the terms.

   Justification

   Algebraic expansion.

   Shown in the video
4. Expression

   $$\lambda^2 - 22\lambda + 96 = 0$$
   Explanation

   Combine constant terms ($121 - 25 = 96$).

   Justification

   Arithmetic simplification.

   Shown in the video
5. Expression

   $$(\lambda - 16)(\lambda - 6) = 0$$
   Explanation

   Factor the quadratic equation.

   Justification

   Factoring technique finding roots 16 and 6.

   Shown in the video
6. Expression

   $$\lambda = 16 \text{ or } \lambda = 6$$
   Explanation

   Solve for lambda.

   Justification

   Zero product property.

   Shown in the video

The eigenvalues of AA^T are 16 and 6.

1. Formula
   Observation

   Screen shows sigma_1 = sqrt(16) = 4 and sigma_2 = sqrt(6).

1. Expression

   $$\sigma_i = \sqrt{\lambda_i}$$
   Explanation

   Apply the definition relating singular values to eigenvalues.

   Justification

   Definition of singular values.

   Shown in the video
2. Expression

   $$\sigma_1 = \sqrt{16} = 4$$
   Explanation

   Substitute the first eigenvalue lambda_1 = 16.

   Justification

   Given eigenvalue value.

   Shown in the video
3. Expression

   $$\sigma_2 = \sqrt{6}$$
   Explanation

   Substitute the second eigenvalue lambda_2 = 6.

   Justification

   Given eigenvalue value.

   Shown in the video

The singular values of A are 4 and sqrt(6).

1. Formula
   Observation

   The slide introduces Example with $A = [[1,-1,3],[3,1,1]]$ and asks to find the singular values of A.

2. Audio
   Observation

   Narrator says "Let's take a look at an example" and proceeds through transpose, products, and eigenvalues.

1. The clip ends after finding eigenvalues 16 and 6; it does not display the final square-root step explicitly.

Let $A = [[1, -1, 3], [3, 1, 1]]$. Find the singular values of A.

1. A is the 2 x 3 matrix [[1, -1, 3], [3, 1, 1]].

2. Singular values are square roots of positive eigenvalues of $A^T A$ or AA^T.

3. $A^T A$ and AA^T have the same positive eigenvalues.

Reduce the problem to finding eigenvalues of the smaller Gram matrix and compute those eigenvalues.

1. Expression

   $$A^T=\begin{bmatrix}1&3\\-1&1\\3&1\end{bmatrix}$$
   Explanation

   Transpose A by converting rows to columns.

   Justification

   Definition of transpose.

   Shown in the video
2. Expression

   $$A^TA=\begin{bmatrix}10&2&6\\2&2&-2\\6&-2&10\end{bmatrix}$$
   Explanation

   Compute the 3 x 3 product $A^T A$.

   Justification

   Matrix multiplication.

   Shown in the video
3. Expression

   $$AA^T=\begin{bmatrix}11&5\\5&11\end{bmatrix}$$
   Explanation

   Compute the 2 x 2 product AA^T.

   Justification

   Matrix multiplication.

   Shown in the video
4. Expression

   $$\text{Choose }AA^T\text{ because it is }2\times2.$$
   Explanation

   Select the smaller Gram matrix for the eigenvalue calculation.

   Justification

   Stated strategy in the video, based on equal positive eigenvalues.

   Shown in the video
5. Expression

   $$\det\left(\begin{bmatrix}11-\lambda&5\\5&11-\lambda\end{bmatrix}\right)=0$$
   Explanation

   Write the characteristic equation for AA^T.

   Justification

   Eigenvalue definition.

   Shown in the video
6. Expression

   $$(11-\lambda)^2-25=0$$
   Explanation

   Evaluate the 2 x 2 determinant.

   Justification

   2 x 2 determinant formula.

   Shown in the video
7. Expression

   $$\lambda^2-22\lambda+96=0$$
   Explanation

   Expand and simplify to a quadratic.

   Justification

   Algebra.

   Shown in the video
8. Expression

   $$(\lambda-16)(\lambda-6)=0$$
   Explanation

   Factor the quadratic.

   Justification

   Factoring.

   Shown in the video
9. Expression

   $$\lambda=16\quad\text{or}\quad \lambda=6$$
   Explanation

   Obtain the two eigenvalues of AA^T.

   Justification

   Zero-product property.

   Shown in the video

The example reaches eigenvalues 16 and 6 of AA^T. By the definition stated earlier, the singular values would be their positive square roots, but that final square-root line is not shown within this clip.

Substitute lambda=16 and lambda=6 into lambda^2-22lambda+$96=0$; both give 0, confirming they are roots of the characteristic polynomial.

1. Formula
   Observation

   Full problem setup and solution displayed on slides.

Determine the singular values of the matrix $A = [[1, -1, 3], [3, 1, 1]]$.

1. $A = \begin{bmatrix} 1 & -1 & 3 \\ 3 & 1 & 1 \end{bmatrix}$

Find sigma_1 and sigma_2.

1. Expression

   $$AA^T = \begin{bmatrix} 11 & 5 \\ 5 & 11 \end{bmatrix}$$
   Explanation

   Calculate the product of A and its transpose.

   Justification

   Matrix multiplication rules.

   Shown in the video
2. Expression

   $$\det(AA^T - \lambda I) = 0 \Rightarrow \lambda^2 - 22\lambda + 96 = 0$$
   Explanation

   Form and simplify the characteristic equation.

   Justification

   Eigenvalue definition.

   Shown in the video
3. Expression

   $$\lambda_1 = 16, \lambda_2 = 6$$
   Explanation

   Solve the quadratic equation for eigenvalues.

   Justification

   Factoring.

   Shown in the video
4. Expression

   $$\sigma_1 = \sqrt{16} = 4, \sigma_2 = \sqrt{6}$$
   Explanation

   Take the square root of each positive eigenvalue.

   Justification

   Singular value definition.

   Shown in the video

The singular values are sigma_1 = 4 and sigma_2 = sqrt(6).

Both eigenvalues were positive, satisfying the condition for real singular values.

1. Animation
   Observation

   The screen changes from the first slide containing the definition and example setup to a new slide titled around finding eigenvalues of AA^T.

1. First slide with definition and example matrices

2. Second slide with eigenvalue recall and characteristic equation

1. Content shifts from computing $A^T$, $A^T A$, and AA^T to recalling the eigenvalue definition and solving det(AA^T - lambda I)=0.

1. The topic remains singular values of the same example matrix A.

2. The chosen matrix for eigenvalues remains AA^T.

The visual transition marks the move from constructing the Gram matrices to solving the eigenvalue problem for the smaller one.

1. Animation
   Observation

   A yellow cursor/highlight moves across formulas as the narrator reads them, including the definition lines, matrix products, and the determinant expansion.

1. Yellow cursor/highlight

2. Displayed formulas and matrices

1. The highlight moves from the definition bullets to the example matrix, then to $A^T$, $A^T A$, AA^T, and finally the characteristic equation steps.

1. The underlying slide content stays static except for newly revealed lines.

The moving highlight synchronizes spoken explanation with the exact formula being discussed, helping identify which matrix or algebraic step is active.

1. Animation
   Observation

   The slide content changes from the eigenvalue calculation to the singular value definition and final results.

1. Previous slide with determinant calculations

2. New slide with sigma formulas

1. Content replacement

1. Title bar 'Singular Values of a Matrix'

Marks the shift from finding intermediate eigenvalues to computing the final singular values.

1. Animation
   Observation

   A grey box with text 'Reminder: Only use the positive eigenvalues...' fades in on the right side.

1. Grey reminder box

1. Box appears

1. Main calculation area

Highlights a crucial constraint in the definition of singular values.

1. Formula
   Observation

   The definition explicitly says "square roots of the positive eigenvalues".

2. Audio
   Observation

   Narrator repeats "positive eigenvalues" twice.

One might think any eigenvalue of $A^T A$ or AA^T can be square-rooted to get singular values.

The video specifically restricts the definition to positive eigenvalues.

1. Formula
   Observation

   Bullet states "$A^T A$ and AA^T have the same positive eigenvalues."

One might assume $A^T A$ and AA^T produce unrelated eigenvalue lists.

The lesson states that their positive eigenvalues are the same, so either product may be used for singular values.

1. Audio
   Observation

   Narrator chooses AA^T because it is 2 x 2 instead of 3 x 3.

2. Formula
   Observation

   Slide text: "Let's find the eigenvalues of AA^T since AA^T is a 2 by 2 matrix."

One might think the standard route is always to use $A^T A$ regardless of size.

The video demonstrates choosing the smaller Gram matrix to simplify the eigenvalue computation.

1. Caption evidence
   Observation

   Text explicitly warns: 'Only use the positive eigenvalues to find the singular values.'

Students might attempt to take the square root of negative eigenvalues when calculating singular values.

Singular values are defined as the square roots of the *positive* eigenvalues of AA^T (or A^TA). Since AA^T is positive semi-definite, its eigenvalues are non-negative, but zero eigenvalues yield zero singular values, and negative ones (if arising from numerical error or wrong matrix) are not used for real singular values.

1. Formula
   Observation

   Definition line introduces singular values via eigenvalues of $A^T A$ or AA^T.

The equality of positive eigenvalues justifies choosing either $A^T A$ or AA^T in the definition.

1. Audio
   Observation

   Narrator applies the transpose rule directly in the worked example.

The example begins by computing $A^T$ using the transpose method.

1. Formula
   Observation

   Both $A^T A$ and AA^T are displayed before the slide states to use AA^T because it is 2 x 2.

Choosing the smaller Gram matrix is valid because the two products have the same positive eigenvalues.

1. Formula
   Observation

   The recall line is immediately followed by det(AA^T - lambda I)=0 and its simplification.

The characteristic-equation derivation uses the recalled definition of eigenvalues.

1. Formula
   Observation

   The example matrix AA^$T = [[11,5],[5,11]]$ is substituted into the eigenvalue equation.

The worked example is carried out by applying the eigenvalue derivation to AA^T.

1. Formula
   Observation

   Formula sigma_i = sqrt(lambda_i) links the two concepts directly.

Finding the eigenvalues of AA^T is a necessary prerequisite step before calculating the singular values of A.

1. Formula
   Observation

   The calculation of AA^T precedes the determinant setup.

Computing the matrix product AA^T is required to form the matrix for which eigenvalues are calculated.

1. Formula
   Observation

   Opening definition slide.

1. Definition of singular values via positive eigenvalues
2. Equality of positive eigenvalues of $A^T A$ and AA^T

1. Audio
   Observation

   Narrator explains choosing AA^T because it is 2 x 2.

1. Choosing the smaller Gram matrix for eigenvalues
2. Equality of positive eigenvalues of $A^T A$ and AA^T

1. Formula
   Observation

   Characteristic equation and factorization are shown step by step.

1. Eigenvalue definition via characteristic determinant
2. Find the eigenvalues of AA^T by solving its characteristic equation

1. Audio
   Observation

   Narrator states rows become columns.

1. Computing the transpose of a matrix

1. Formula
   Observation

   The clip ends at lambda = 16 or lambda = 6 without showing the square-root conclusion.

1. Final numeric singular values are not explicitly displayed in this clip.

1. Definition of singular values via positive eigenvalues
2. Worked example: set up singular values for a 2 x 3 matrix

1. Audio
   Observation

   Narrator explains the process of taking square roots of eigenvalues.

1. Definition of Singular Values via Eigenvalues
2. Calculation of Singular Values from Eigenvalues

1. Caption evidence
   Observation

   Reminder box about positive eigenvalues.

1. Using Negative Eigenvalues for Singular Values
2. Constraint on Eigenvalues for Singular Values

1. Formula
   Observation

   Step-by-step algebraic expansion shown.

1. Characteristic Equation for Eigenvalues
2. Derivation of Eigenvalues from Characteristic Polynomial