4. Matrices Flashcards by owen Knowles

For an element Amn, what does the m and n determine

M refers to the row of the element (height- think mountain)
N refers to column of the element (length- think Nile)

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What is a vector?

A matrix with only one row or column

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What is the trace?

It is the sum of the elements along the diagonal of a square matrix

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Diagonal matrices

Square matrices with non zero entries only in the main diagonal and zero entries everywhere else

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Identity matrix

A special type of diagonal matrix (In) with the number 1 in the main diagonal and zero everywhere else

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Triangular matrix

Non zero entries in the positions above (below) the main diagonal and zero entries below (above) the main diagonal

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Scalar

A matrix with a single entry (ie a real number)

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When can you add and subtract matrices?

When they are of the same order

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When can you multiply matrix Amxn and matrix Bpxq? And what dimensions will the new matrix have?

When n=p
The new matrix will have order m x q

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What does B x In equal when B is a matrix?

B
Any matrix times by the identity matrix equals itself

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What is a matrix transposition?

The rows and columns of the matrix are interchanged. It is essentially reflected in the line y=-x

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Whet is the inverse matrix?

It is the matrix which satisfies the condition A x A^-1= A^-1A=In

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What is 1/|A| x adj(A) equal to?

The inverse matrix A^-1

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The determinant

|A| is a number which is a function of the entries of a square matrix

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When can an inverse matrix be determined?

Only for a square matrix. Only if the determinant of the matrix doesn’t equal zero

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What is the determinant of a matrix of order 1?

a11 (the only element available)

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How is the determinant calculated?

By multiplying it’s elements by the corresponding cofactors and adding the products

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Adjoint matrix

The transpose of the matrix obtained by replacing each element aij of matrix A with its corresponding cofactor

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Does (AB)’ = B’A’

Yes

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When is a matrix called singular?

When it doesn’t have an inverse

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When are the inverses of matrices unique?

Always

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What is (AB)^-1 equal to? Assuming AB is invertible

B^-1 x A^-1

What is (cA)^-1 equal to assuming c is a number not equal to 0

c^-1 x A^-1

What is the determinant used for?

To find the inverse matrix or determine if it exists

What is the sarrus rule?

A way of finding the determinant in a 3x3 matrix |A|= a11a22a33 + a12a23a31 + a13a21a32- a31a22a13- a32a23a11- a33a21a12

What is the minor of an element

|Aij| with deleted row i and column j is called the minor of element aij

What is the cofactor of an element?

The cofactor is the minor with the appropriate sign Cij= (-1)^(i+j) |Aij|

What is |A’| equal to?

|A|

What is |AB| equal to?

|A| x |B|

What is the way of finding the inverse of a particular element in a matrix?

Aij^-1= 1/|A| x Cij

What can be said about the equation system Ax=b if |A| is not equal to zero?

There is an unique inverse A^-1. The solution x=A^-1 x b is the only solution

When dealing with eigenvalues what is k?

An unknown n element column vector

What is r

An unknown scalar

When do we get the trivial solution k=0

When A-rI is non singular

When do we get a non trivial solution to k?

When A-rI is singular so the determinant of A-rI=0

Eigenvalues

The values of r that are a solution to det(A-rI)=0

Eigenvector

A nonzero vector ki which is a particular solution of the equation for a particular eigenvalue ri

Whet does the trace of matrix A equal?

The sum of the eigenvalues of A

What does the product of the eigenvalues of A equal?

The determinant of A

When is a quadratic form q(x) positive definite?

If q(x)>0 for all x=\ 0

When is a quadratic form q(x) positive semi definite?

If q(x)>= for all x =\0

When is a quadratic form q(x) negative definite?

If q(x)<0 for all x =\0

When is a quadratic form q(x) negative semi definite?

If q(x)<=0 for all x=\0

How can whether a quadratic form is +ve or -ve definite be checked

By looking at its leading principal minors I.e. the determinants of the leading principal is matrices of A

When is a symmetric matrix positive definite?

If every leading principal minor is positive

When is a symmetric matrix negative definite?

If the leading principal minors alternate in sign starting with negative

When is a symmetric matrix positive semi definite?

Determinants of all principal submatrices >=0

When is a symmetric matrix negative semi definite

Determinants of all principal submatrices of odd order <=0 and determinants of all principal submatrices of even order >=0

Using eigenvalues when is a symmetric matrix +ve definite?

If all eigenvalues are +ve

Using eigenvalues when is a symmetric matrix +ve semidefinite?

If all eigenvalues are non -ve

Using eigenvalues when is a symmetric matrix -ve definite?

If all eigenvalues are -ve

Using eigenvalues when is a symmetric matrix -ve semidefinite?

If all eigenvalues are non positive

Using eigenvalues when is a symmetric matrix indefinite?

If it has a +ve and -ve eigenvalues