Matrix Operations - Aerospace Engineering Courses …aeweb.tamu.edu/haisler/engr214/Word_Lecture_Notes_by... · Web viewMatrix Operations Matrix - a collection of numbers or other

2001, W.E. Haisler Introduction to Matrix Algebra and Vector Mechanics

Matrix Operations

Matrix - a collection of numbers or other items arranged in a particular manner in an array.

Rectangular Matrix - array with n rows and m columns)

= term in row i and column j

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Column Matrix – only 1 column in the matrix (sometimes called a column vector)

or

Row Matrix – only 1 row in the matrix (sometimes called a row vector)

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Matrix Addition

[A] and [B] must be the same size!

Matrix Multiplication

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The number of columns in [A] (i.e., p) must be equal to the number of rows in [B].


The definition of is equivalent to taking the dot product of row i of matrix [A] and column j of matrix [B].

If [A] is (2x4) and [B] is (4x3):

and

then,

Pictorially, called “row into column multiplication”

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Note: In general, matrix multiplication is not commutative, i.e.,

Matrix Division – NOT DEFINED! Instead the matrix inverse is defined. If where [I] is the identity matrix (all zeros except for 1’s on main diagonal running from upper left to lower right), then such that .

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Transpose Matrix – Given [A], then the transpose is given by and is formed by interchanging all rows with corresponding columns. An (nxm) matrix becomes an (mxn) matrix.

Symmetric Matrix – A matrix is symmetric about its main diagonal (diagonal running from upper left to lower right) if

. For example, for a (3x3), we have 9 values, but the 3 values below the diagonal are equal to the corresponding 3 values above the diagonal; hence, only 6 unique values due to the symmetry.

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System of n Equations –

or, in matrix notation:

or [A]{X}={C}

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Vector Operations & Operators

Scalar Product (also called dot or inner product)a. Definitions : Given: two vectors and ,

b. Observations 1) Vector Vector Scalar (one order down from a

vector)2) Vector Matrix (second order tensor) Vector

(one order down from a second order tensor)3) The dot product ALONE is commutative &

distributive

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Note: if one of the quantities in a dot product has differentiation in it, the commutative property of the dot product will not hold.

4) ADVANTAGE of 2nd definition of dot product? Do not have to evaluate magnitude of vectors, i.e. do not have to calculate the following:

5) Physical meaning? Projection of one directional quantity (vector, second order tensor, or higher tensor) on to another directional quantity (vector, second order tensor, or higher tensor)

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6) USES:a. Find the angle between two vectorsb. Find the magnitude of the projection of one vector onto another (parallel component)c. Determines orthogonality (dot product = zero, then orthogonal)

What does really mean in terms of how it is evaluated? It is very similar to algebraic multiplication. For example, if you have the algebraic product , you get 9 terms in the product:

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If you have the vector product

you still get nine terms but they include the unit vectors and the dot product operator:

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Cross Producta. Definition:

b. Alternate approach: (a scalar not a vector, direction comes from right hand rule)

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c. Observations1)Physical meaning? is normal () to

the plane defined by

2)Commutative property does not hold

3)Distributive property does hold

4)USES:a. Magnitude of Cross Product is the area of the parallelogram mapped by the two vectorsb. Calculates Moments:

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Dyadic Product (also called outer, tensor, or vector product)a. Definition :

This is called a Tensor because each term has two unit vectors.Notice: Two Unit Vectors accompany each entry. First Unit Vector denotes face (on a cube). Second Unit Vector denotes direction of vector.

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The dyadic product means that you are multiplying the 3 components of times the 3 components of

(like algebra) and then arranging results into a (3x3) matrix.


b. Observations 1) No “vector operator” between vectors (though

sometimes the symbol is used, i.e., )2) VECTOR times VECTOR Matrix (2nd Order

Tensor)3) Note matrix representation: {3x1}[1x3] gives a

[3x3] by distributing each component of the first vector over the second vector to form the three “rows” of the matrix representation of the second order tensor

4) USES:a. Stress is a Tensorb. Strain is a Tensor

c. Dot products between vectors and second order tensors is given by

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The above dot product means that 9 quantities are dotted with 3 quantities.

You get 27 terms. However, 18 of these disappear (because , etc.), so only 9 terms are left.

Arrange 9 terms as a (3x3 matrix).

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1) Vector operation is just like a matrix operation in this case, i.e., calculations are done by “row down column” method which yields a vector

2) E.g. #1 (Second Unit Vector of Tensor Dots with Unit Vector of Vector leaving First Unit Vector of Tensor to form the New Vector)

3) E.g. #2 (Unit Vector of Vector Dots with First Unit Vector of Tensor leaving Second Unit Vector of Tensor to form the New Vector)

Suppose you have instead. Using same procedure to take the dot product, you obtain:

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Note that a tensor is denoted by two arrows above it, i.e., . While the dot product of a vector and a vector yields a scalar, the dot product of a vector and a tensor yields a vector.

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Differential Operators

1. Derivatives (Review of Calculus I)a. Total derivative: Given:

Physical meaning? Slope of line

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b. Partial Derivatives 1) Given:

(holding y constant)

(holding x constant)

2) Example:

3) Physically, slope with respect to x at a fixed y

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2. Del Operator (Review of Calculus II and III)

a. Definition:

i.e., “A differential vector” in Cartesian coordinates

b. Divergence (of B):

Note that yields a completely different result:

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The above is a vector operator just like is a vector operator.

Hence, when you take the dot product of two vector AND one of the vectors is an operator (like ), you cannot interchange the order of operation (like you can with a simple dot product).

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c. Curl:

d. Gradient for :

e. LaPlacian:

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ASIDE: Review of Unit Vectors

A. Definition: a vector whose magnitude is one.Given:

Then: is a unit vector if: B. Examples:

1. Is this a unit vector:

2. Is this a unit vector:

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3. How would you make a unit vector? Divide by it’s magnitude:

The symbol “^” is sometimes used over a vector to denote a unit vector.

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Determining normal to a plane

Recall equation of a plane:

Define :

Gradient of gives a normal vector to the plane:

Divide by the magnitude to get a unit normal:

Problem! You don’t know which direction the unit normal points (into or out of plane surface).

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Another approach. Define two vectors and as shown. These vectors obviously lie in the plane of the surface. Do . This gives a vector perpendicular to and (and the normal points outward because of right hand rule!), hence is perpendicular to the plane. The

unit normal to the surface is then given by .

To find the component of vector perpendicular to the surface (i.e., in direction of , do . The vector is then given by

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Angle between and can be found from

or

Important note. In the above, and on the following page, when you determine the component of a vector (say ) in the direction of another vector (say ) using the dot product, THE VECTOR MUST BE A UNIT VECTOR.

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Determining components of a vector that are normal and parallel to a surface with unit vector normal to the surface.

The normal component is first obtained from the dot product

or as a vector The parallel component is obtained from vector addition

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Matrix Operations - Aerospace Engineering Courses …aeweb.tamu.edu/haisler/engr214/Word_Lecture_Notes_by... · Web viewMatrix Operations Matrix - a collection of numbers or other