emth211 concepts

What does an eigenvector mean?

It is a nonzero vector whose direction is only scaled by the matrix: A v equals lambda v. The scaling factor lambda is its eigenvalue.

What is algebraic multiplicity?

The number of times an eigenvalue appears as a root of the characteristic polynomial which is det(A minus lambda I).

What is geometric multiplicity?

The dimension of the eigenspace: the maximum number of independent eigenvectors for that eigenvalue. It is between 1 and its algebraic multiplicity.

When is an n by n matrix diagonalisable?

When it has n independent eigenvectors over the chosen field. Equivalently all eigenvalues lie in that field and each has equal geometric and algebraic multiplicities.

Does a repeated eigenvalue prevent diagonalisation?

Not necessarily. It prevents diagonalisation when its geometric multiplicity is smaller than its algebraic multiplicity. Such a matrix is defective.

Do n distinct eigenvalues guarantee diagonalisation?

Yes over a field containing those eigenvalues because eigenvectors for distinct eigenvalues are independent.

What is trace and how is it related to eigenvalues?

The sum of diagonal entries. It also equals the sum of eigenvalues counted with algebraic multiplicity even for a defective matrix.

How is determinant related to eigenvalues?

It is their product counted with algebraic multiplicity even for a defective matrix.

When is a square matrix invertible?

Exactly when zero is not an eigenvalue. Equivalently its determinant is nonzero and its columns are independent. Diagonalisability is a separate property.

What does the Cayley-Hamilton theorem mean?

Every square matrix satisfies its own characteristic equation: p(A) equals the zero matrix. This includes defective matrices. A constant term c becomes c times I.

Why is Cayley-Hamilton useful?

It lets high powers be expressed using lower powers. For an invertible matrix it also lets the inverse be expressed as a polynomial in A.

What does the spectral theorem say about a real symmetric matrix?

It has real eigenvalues and an orthonormal eigenvector basis: A equals Q D Q transpose. Real symmetric matrices are exactly those that are orthogonally diagonalisable.

What do P and D represent in a diagonalisation?

In A equals P D P inverse the columns of P are independent eigenvectors and the diagonal entries of D are their matching eigenvalues.

What makes an eigenvalue dominant in this course?

Its absolute value is strictly greater than that of every other eigenvalue. It need not be the largest positive eigenvalue.

What makes a transition matrix stochastic in this course?

Its entries are nonnegative and every column sums to one. Entry a i j is the probability of moving from state j to state i.

What is a stationary probability distribution?

A nonnegative vector with entries summing to one that satisfies A v equals v. It stays unchanged after a transition.

What does regular mean for a stochastic matrix?

Some positive integer power of the matrix has every entry strictly positive.

What does regularity guarantee for a Markov chain?

Every initial probability distribution approaches the same unique stationary distribution. Eigenvalue 1 is simple and every other eigenvalue has absolute value below one.

What do the dominant eigenvalue and eigenvector mean in a Leslie model?

When a stable age structure exists the eigenvalue gives the long-term growth factor per step and the eigenvector gives age proportions. Above one means growth; below one means decline.

What does Gershgorin's theorem guarantee?

Every eigenvalue lies in the union of the row discs and also in the union of the column discs. Individual overlapping discs may contain no eigenvalues.

What does a separated group of k Gershgorin discs contain?

Exactly k eigenvalues counted with algebraic multiplicity if separated from the remaining discs in the same row or column family. An isolated disc contains exactly one.

What can Gershgorin discs tell you about invertibility?

If zero is excluded from the allowed region the matrix is invertible. If zero is included that alone does not prove singularity.

What restriction do complex eigenvalues place on a real matrix?

Nonreal eigenvalues come in conjugate pairs. Therefore the number of real eigenvalues counted with multiplicity has the same odd or even parity as the matrix size.

What properties define a real inner product?

Symmetry and additivity and scalar homogeneity and positive definiteness: the self inner product is nonnegative and equals zero only for the zero vector.

What do norm and orthogonality mean in an inner product space?

Norm is the square root of the self inner product. Two vectors are orthogonal when their inner product is zero.

What is the goal of least squares?

Choose coefficients that minimise the squared length of the residual: the difference between the data vector and the fitted vector.

What does the best approximation theorem say?

The orthogonal projection onto a finite-dimensional subspace is the unique vector in that subspace closest to the original vector.

What geometric property characterises a least squares solution?

The fitted vector is the projection of b onto the column space of A. The residual is orthogonal to that column space.

When are least squares coefficients unique?

Exactly when A has linearly independent columns. Even without unique coefficients the fitted vector is always unique.

Which real matrices have a singular value decomposition?

Every real matrix including rectangular and rank-deficient matrices: A equals U Sigma V transpose with orthogonal U and V and nonnegative singular values.

What are singular values and how do they differ from eigenvalues?

They are nonnegative scaling factors from the SVD. Their squares are eigenvalues of A transpose A. For symmetric A they are absolute values of its eigenvalues.

How does the SVD reveal rank?

Rank equals the number of positive singular values in exact arithmetic. Zero singular values correspond to lost directions.

What does the pseudoinverse give?

A+ times b is the unique least squares solution with smallest Euclidean norm. For a square invertible matrix the pseudoinverse equals the ordinary inverse.

What does R squared mean?

For least squares with an intercept and a nonconstant response it is the fraction of observed response variation explained by the model. It does not establish causation.

What does a coefficient mean in multiple regression?

The change in expected response for a one-unit increase in that predictor while all other included predictors are held fixed.

What is multicollinearity and why does it matter?

Predictors are linearly dependent or nearly dependent. Exact dependence prevents unique coefficients; near dependence can make coefficient estimates unstable.

What does a significant Haitovsky test support in this course?

Evidence against multicollinearity. Failing to reject does not prove multicollinearity is present.

What does a 95 percent confidence interval mean?

Across repeated samples about 95 percent of intervals from that procedure contain the fixed true parameter. It does not describe 95 percent of individual observations.

What is a p value?

Assuming the null hypothesis and model are true it is the probability of a test statistic at least as extreme as observed. It is not the probability the null is true.

What does a slope confidence interval containing zero tell you?

The matching two-sided test does not reject a zero slope at the corresponding significance level. It does not prove the slope is zero.

Why is a prediction interval wider than a mean-response confidence interval?

It includes uncertainty in the fitted mean plus the additional random variation of a new individual response.

What is the difference between standard deviation and standard error?

Standard deviation measures the spread of individual observations. Standard error measures the sampling variability of an estimator such as the sample mean.

What does correlation measure and what does zero correlation mean?

Correlation measures linear association. Zero correlation means no linear association but does not rule out a nonlinear relationship or imply independence.

When should a confidence interval for a population mean use z or t?

For an independent sample from a normal population use z when population standard deviation is known and t with n minus 1 degrees of freedom when it is estimated.

What assumptions justify the usual exact t inference for regression coefficients?

A correct linear mean model with independent normal errors of mean zero and constant positive variance. The design matrix (containing predictor columns) has independent columns and more observations than fitted coefficients.

What does the overall regression F test establish?

It tests whether all predictor slopes are zero with the intercept unrestricted. Rejection supports at least one nonzero slope but does not identify which.

What do the first and later principal components represent?

The first is the unit direction of greatest variance in centred data. Each later direction maximises variance while remaining orthogonal to earlier directions.

What is the difference between a PCA loading and a score?

A loading is a direction in variable space. A score is an observation's coordinate along that direction after subtracting the data mean.

What do covariance eigenvalues tell you in PCA?

Each is the variance of its component's scores. Dividing it by the sum of all covariance eigenvalues gives its explained-variance fraction when total variance is positive.

How does the first PCA line differ from ordinary regression of y on x?

The PCA line minimises squared perpendicular distances. The regression line minimises squared vertical response residuals.