Lecture 6_Matrix algebra

Question 1

What is a matrix?

Answer 1

a two-dimensional ordered array of numbers

Question 2

What are the numbers within a matrix called?

Answer 2

elements

Question 3

How are elements in a matrix identified?

Answer 3

(Row #) then (Column #)

Question 4

What is the order of a matrix?

Answer 4

its size, expressed as the number of its rows and columns.

Question 5

Matrix Addition & Subtraction

Answer 5

Proceed on an element by element basis

Note: two matrices must be of the same order to permit addition or subtraction

Question 6

Scalar multiplication

Answer 6

that each element of a matrix is multiplied by the value of the scalar

Question 7

What conditions must be met for Matrix Multiplication?

Answer 7

the number of columns in the first matrix must equal the number of rows in the second matrix.

Question 8

What are the dimensions of the resulting product matrix from matrix multiplication?

Answer 8

The resulting product matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

Question 9

Is matrix multiplication commutative?

Answer 9

No (A̅ x B̅) ≠ (B̅ x A̅)

Question 10

What does it mean to transpose a matrix?

Answer 10

exchange the rows and columns

Question 11

What are some uses for the transpose of a matrix?

Answer 11

Pre-multiplying D̅ by D̅’ (D̅’ x D̅) gives the Sum of Cross-products matrix for variables X and Y (dividing all elements by sample size gives the variance-covariance matrix)

Question 12

Matrix division

Answer 12

• no operation directly comparable
• must calculate the inverse [B̅÷ A has to be performed as A̅¯¹B̅ (matrix B is pre-multiplied by A̅¯¹, the inverse of matrix A̅)]

Question 13

How do you find the inverse of a matrix?

Answer 13

1st - find the determinant (magnitude) [if = 0 → no inverse]

2nd - use determinant as a scalar in the formula (given) to find inverse

Question 14

What is an Identity matrix?

Answer 14

the result of a matrix multiplied by its inverse

• its diagonal elements are 1s and its off-diagonal elements are all 0s.

Question 15

How do we obtain the standardized coefficients (βᵢ) and R² from the correlation matrix among the variables.

Answer 15

1. find the inverse R̅ᵢᵢ¯¹
2. check by multiplying R̅ᵢᵢ x R̅ᵢᵢ¯¹ → identity matrix
3. calculate βᵢ by B̅ᵢ = R̅ᵢᵢ¯¹ x R̅ᵢy
4. calculate R² = R̅ᵢy x B̅ᵢ
5. calculate bᵢ = βᵢ (SDy/SDᵢ)

Question 16

Working with the diagonal elements of the inverse the correlation matrix among the IVs, we can get the …

Answer 16

standard errors of the regression coefficients (see equation)

Question 17

A Correlation Matrix …

Answer 17

Summarizes the Bivariate Relations Within a System of 3 or more Variables

Question 18

What constraints ensure a stable correlation matrix?

Answer 18

• all correlations must be based on the same sample of observations.
• This requires “listwise” deletion, not “pairwise” deletion, when calculating a correlation matrix.
• ensures that the range of values for correlation in the matrix is constrained by the values of all the other correlations within the system.

Question 19

Always use ___ deletion so that each correlation in
the matrix is based on the same sample (number) of
subjects, even if this means discarding some subjects.

Answer 19

listwise (ensures correlations are constrained)

Question 20

Pairwise deletion can produce a matrix that is ___?

Answer 20

unstable (the internal constraints among correlations do not hold)
- can produce a negative value for the determinant

Question 21

If you use pairwise deletion, what is the actual

sample size used in your study?

Answer 21

unknown (different variables may have different numbers)