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Matrices

Numbers

A single number in matrix notation is called a scalar. It canbe looked at as a number, or as a 1 x 1 matrix, or as a one

element row or column.

Rows

A row(also called a row vector) is just an ordered

collection of elements. For example,

[ a b c ]

is a row.

If you have two rows of the same length, you can add the

rows by adding the corresponding elements in each row.

For example, the row

[ d e f ] + [ g h i ] = [ d+g e+h f+i ]

One can multiply a row by a scalar (number). For example,

2 * [ a b c ] = [ 2a 2b 2c ]

A row may have any number of elements, from one on up.

If Z is a row, Z(i) means the i'th element of that row.

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Assoc. Prof. Dr. Norshuhada Shiratuddin - Lecture-Notes 1

Columns

A column(also called a column vector) is just like a row,

except it is arranged vertically. For example:

[ a ]

[ b ]

[ c ]

Columns can be added, or multiplied by a scalar (number)

the same way that rows can:

[ a ] [ g ] [ a+g ]

[ b ] + [ h ] = [ b+h ][ c ] [ i ] [ c+i ]

[ a ] [ 2a ]

2 * [ b ] = [ 2b ][ c ] [ 2c ]

A column may have any number of elements, from one onup.

If Y is a column, then Y(i) means the i'th element of thecolumn, counting from the top.

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Operations on rows and columns

If you have a row and a column, you can form what's called

an "inner product", or "vector product", or a " dot

product". The inner product is the product of the firstelement of the row with first element of the column, plusthe product of the second two elements, etc. For example:

[ a b c ] [ d ] = [ a*d + b*e + c*f ]

(or just the scalar) a*d + b*e + c*f

[ e ][ f ]

A row and column that have a dot product of zero arecalled Orthogonal vectors .

There is also another kind of product (" outer product")

[ a ] [ d e f ] = [ a*d a*e a*f ][ b ] [ b*d b*e b*f ]

[ c ] [ c*d c*e c*f ]

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Matrices

A matrixcan be looked at as a column of rows, or as a row

of columns.

What is a Matrix?

A matrixis a rectangular collection of like objects, usually

numbers. We are primarily interested in matrices because

they can be used to solve systems of linear equations.

The orderof a matrix is the number of rows and columns.

This is a (3x4) matrix because it contains 3 rows and 4columns.

We denote that the objects are part of a matrix by using

square brackets. An element of the matrix is referred to bytwo subscripts - the first is the row and the second is the

column, ar

c

.

Examples of matrices (with their sizes):

[ a b c ] This is a 3 x 3 matrix.

[ d e f ]

[ g h i ]

[ a b c ] This is a 2 x 3 matrix.

[ d e f ]

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[ a b ] This is a 3 x 2 matrix.[ c d ]

[ e f ]

[ a ] This is a 1 x 1 matrix. Orit is a number, your choice.

(Or it is a scalar)

[ a b c ] This is a 1 x 3 matrix, or

it is a row, also your

choice.

[ a ] 3 by 1 matrix

[ b ][ c ]

If X is a matrix, X(i,j) means the element at the i'th row,

and j'th column of the matrix.

A matrix is called a square matrix if it has the samenumbers of rows as columns.

A zero matrix is a matrix where all the elements are zeros.

For example

[ 0 0 0 ] is a zero matrix.

[ 0 0 0 ][ 0 0 0 ]

A zero matrix serves many of the same functions in matrix

arithmetic that 0 does in regular arithmetic.

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The identity matrix is a matrix that has one's on the

diagonal, and zeros everywhere else. For example:

[ 1 0 0 ] is an identity matrix.

[ 0 1 0 ][ 0 0 1 ]

The identity matrix is usually written "I".

An identity matrix serves may of the same functions in

matrix arithmetic that 1 does in regular arithmetic.

The diagonalof a matrix are the elements that have

identical row and column numbers. (X(i,i)) For example:

The diagonal of [ a b c ] is a, e, andi.

[ d e f ][ g h i ]

A diagonal matrix is one that has non-zero elements only

on the diagonal.

[ * 0 0 0 ] Diagonal matrix.[ 0 * 0 0 ]

[ 0 0 * 0 ]

[ 0 0 0 * ]

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A block diagonal matrix is like a diagonal matrix, except

that elements exist in the positions arranged as blocks.

Example: (Where * means a non-zero element.)

[ * 0 0 0 0 0 0 0 ]

[ 0 * * 0 0 0 0 0 ][ 0 * * 0 0 0 0 0 ]

[ 0 0 0 * * * 0 0 ] B lock diagonal matrix.[ 0 0 0 * * * 0 0 ]

[ 0 0 0 * * * 0 0 ]

[ 0 0 0 0 0 0 * * ][ 0 0 0 0 0 0 * * ]

A Band Matrixhas numbers near the diagonal of the

matrix, and nowhere else. The width of the band is called

the band width of the matrix.

[ * * * 0 0 0 0 0 ]

[ * * * * 0 0 0 0 ][ * * * * * 0 0 0 ]

[ 0 * * * * * 0 0 ] Band matrix.

[ 0 0 * * * * * 0 ]

[ 0 0 0 * * * * * ][ 0 0 0 0 * * * * ]

[ 0 0 0 0 0 * * * ]

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A matrix is called sparseif most of the elements in it are

zero.

[ 0 0 0 * 0 0 0 0 ]

[ 0 * 0 0 0 0 * 0 ][ 0 0 0 0 * 0 0 0 ]

[ 0 0 * 0 0 0 0 0 ] Sparse matrix.

[ * 0 0 0 0 0 0 0 ][ 0 0 0 0 0 0 0 * ]

[ 0 0 * 0 0 * 0 0 ][ 0 0 0 * 0 0 0 0 ]

A matrix is called denseif it is not sparse.

The transpose of a matrix "N" (Written N') is just a matrix"P" such that N(i,j) = P(j,i). For example:

transpose ( [ a b c ] ) = [ a d ]

[ d e f ] [ b e ]

[ c f ](The matrix has just been reflected

across the diagonal)

Or with ' notation:if A = [ a b c ] then A' = [ a d ]

[ d e f ] [ b e ]

[ c f ]

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If a matrix M has the property that

M'M = I

than the matrix is called an Orthogonal matrix

The transpose of a product (AB)' is the product of the

individual transposes in reverse order. (B'A')

The transpose of a row is a column, and the transpose of a

column is a row. Example:

[ a b c]' = [ a ]

[ b ][ c ]

It is usually easier to write column matrices using

transposes as

[a b c]'

than as[a]

[b][c]

If a matrix is equal to its own transpose, it is called a

symmetricmatrix. (See below)

The transpose of a number is just a number.

(AB)' = B'A' in general. (A transpose of a product is the

product of the transposes in reverse order)

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A symmetric matrix is a square matrix equal to it's

transpose. For example:

[ a b c d ] is a symmetric matrix.

[ b e f g ] (Note that it looks like it was reflected[ c f h i ]across the diagonal.)

[ d g i j ]

A 1x1 matrix (just a number) or a scalar is symmetric.

A symmetric matrix has the property that A = A'.

A matrix with entries only below the diagonal, or with

entries only above the diagonal, is called a (lower, upper)

triangular matrix . If the diagonal in those cases consists

only of 1's, then the matrix is unit triangular. Examples:

upper triangular lower triangular

[ * * * * * ] [ * 0 0 0 0 ]

[ 0 * * * * ] [ * * 0 0 0 ][ 0 0 * * * ] [ * * * 0 0 ]

[ 0 0 0 * * ] [ * * * * 0 ]

[ 0 0 0 0 * ] [ * * * * * ]

upper unit triangular lower unit triangular[ 1 * * * * ] [ 1 0 0 0 0 ]

[ 0 1 * * * ] [ * 1 0 0 0 ][ 0 0 1 * * ] [ * * 1 0 0 ]

[ 0 0 0 1 * ] [ * * * 1 0 ]

[ 0 0 0 0 1 ] [ * * * * 1 ]

Matrices of the same size may be added, by making a newmatrix of the same size, with elements that just add the

corresponding elements from the matrices being added. For

example:

[ a b c ] + [ h i j ] = [ a+h b+i c+j ]

[ d e f ] [ k l m ] [ d+k e+l f+m ]

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