Matrix Operations

Definition 1. Matrices of the same shape can be added and subtracted.

Let A and B be r × c matrices with A = [aij] and = [bij]. Then A + B is an r × c matrix with A + B = [aij + bij] and A – B is an r × c matrix with A – B = [aij – bij].

Definition 2: A matrix can be multiplied (or divided) by a scalar. A scalar can also be added to (or subtracted from) a matrix.

Let A be an r × c matrix with A = [aij] and let b be a scalar. Then bA and A + b are r × c matrices where bA = [b · aij] and A + b = [aij b]. We can define Ab and b + A in a similar fashion. Clearly, b + A = A + b and Ab = bA. Division and subtraction of matrices by scalars can be defined similarly.

Definition 3: Two matrices can also be multiplied, but only if they have compatible shape.

Let A be a p × m matrix with A = [aij], and let B be an m × n matrix with B = [bjk]. Then AB is an p × n matrix with AB = [cik] where

Matrix multiplication formula

Observation: For the multiplication AB to be valid, the number of columns in A must equal the number of rows in B. The resulting matrix will have the same number of rows as A and the same number of columns as B.

The associative law holds, namely (AB)C = A(BC), i.e. it doesn’t matter whether you multiply A by B and then multiply the result by C or first multiply B by C and then multiply A by the result. It is essential that the matrices have compatible shape. Thus if A is p × m, B is m × n and C is n × s then ABC will have shape p × s. The distributive laws, namely A(B + C) = AB + BC and (A + B)C = AC + BC, also hold.

The commutative law of addition holds, namely A + B = B + A, but the commutative law of multiplication does not hold even when the matrices have suitable shape; thus, even for two n x n matrices A and B, AB is not necessarily equal to BA. For square matrices the trace of AB is equal to the trace of BA though.

Property 0: For square matrices A and B of the same size and shape and scalar c:

  1. Trace(A+B) = Trace(B+A)
  2. Trace(cA) = c Trace(A)
  3. Trace(AB) = Trace(BA)

Proof: The proofs are straightforward, based on the definition of trace and matrix addition and multiplication.

Definition 4: The transpose of an r × c matrix A = [aij]   is the c  × r matrix AT = [aji].

A (square) matrix A is symmetric if A = AT

Property 1:

  1. (AT)T = A
  2. (AB)T = BTAT
  3. If A and B are symmetric and AB = BA then AB is symmetric
  4. Trace(A) = Trace(AT)

Proof: We prove (c). Assume that A and B are symmetric. By Definition 4 and Property 1b, AB = ATBT = (BA)T = (AB)T

Observation: If A is a column vector then ATA is a scalar. In fact, AT= ‖A‖. Thus, a column vector A is a unit vector if and only if AT= 1.

Definition 5: An n × n matrix A is invertible (also called non-singular) if there is a matrix such that AB = BA = In. A-1 is the inverse of A provided AA-1 = A-1A = In. A matrix which is not invertible is called singular.

Property 2: If A is invertible, then the inverse is unique.

Proof: Suppose B and C are inverses of A. Then by the associative law,  C = IC = (BA)C = B(AC) = BI = B, and so C = B.

Observation: In fact, if there is a matrix  such that AB = In or BA = In then A is invertible and A-1 = B.

Property 3:  If A and B are invertible, then (A-1)-1 = A and (AB)-1 = B-1 A-1

Proof: The first assertion results from the first assertion of Property 2.

Since (AB)(B-1A-1) = A(BB-1)A-1,= AIA-1 = AA-1 = I the second assertion follows from the second assertion of Property 2.

Property 4: If A is invertible, then so is its transpose and (AT)-1 = (A-1)T

Proof: By Property 1b, AT(A-1)T = (A-1A)T = IT = I. Similarly, (A-1)TAT = (AA-1)T = IT = I.

Property 5: A is symmetric if and only if A-1 is also symmetric

Proof: Assume A is symmetric, then by Property 4,  (A-1)T = (AT)-1 = A-1, and so A-1 is also symmetric. For the converse, assume that A-1 is symmetric, then from the above, it follows that (A-1)-1 is symmetric, but by Property 3, this means that A is symmetric.

Example 1: Find the inverse of

image5069

Since the inverse of A takes the form

Matrix inverse

where AA-1 = I2, it follows that

Thus we need to solve the following four linear equations in four unknowns:

image2888 image2889 image2890

Solving these equations yields a = 2/3, b = -1/3, c = 1/3, d=1/3, and so it follows that

image2893

Excel Functions: Excel provides the following array functions to carry out the various matrix operations described above (where we conflate the arrays A and B with the ranges in an Excel worksheet that contain these arrays).

MMULT(A, B): If A is an p × m array and B is an m × n array, then MMULT(A, B) = the p × matrix AB. Note that since this is an array function, you must first highlight a p × n  range before entering =MMULT(A, B) and then you must press Ctrl-Shft-Enter.

MINVERSE(A): If A is an n × n square array, then MINVERSE(A) = A-1. This is an array function and so you must highlight an n × n range before entering =MINVERSE(A) and then pressing Ctrl-Shft-Enter.

TRANSPOSE(A): If A is an m × n array, then TRANSPOSE(A) = AT. This is an array function and so you must highlight an n × m range before entering =TRANSPOSE(A) and then pressing Ctrl-Shft-Enter.

Excel 2013 also provides the array function MUNIT(n) which returns the n × n identity matrix.

You can also transpose an array A in Excel by copying the array (i.e. by highlighting the array and pressing Ctrl-C), clicking where you want AT located (i.e. the cell at the upper left corner of AT) and then selecting Home > Clipboard|Paste and choosing the Transpose option.

In addition, if A and B are defined as arrays (e.g. they are named arrays or entities such as B5:F8 or they are the results of matrix operations such as TRANSPOSE, INVERSE or MMULT, then they can be manipulated using the +, -, *, / and ^ operators. These operations are done on a cell by cell basis.

For example, suppose range B2:C3 contains

image2900

If you highlight range D7:E8, enter =2*B2:C3+TRANSPOSE(B2:C3) and then press Ctrl-Shft-Enter, D7:E8 will contain

image5071

Note that D7:E8 must have the same shape as B2:C3 or an error will result. Note too that

image5073

Note too that if A is an m × n matrix and B is a 1 × n matrix (i.e. a row vector) then A + B is a valid operation in Excel and gives the same result as A + C where C is an m × n matrix all of whose rows contain the same data as B. Similarly you can calculate A – B, A*B and A/B. Also, you can calculate A + B, A – B, A*B and A/B where B is an m × 1 column vector.

E.g. Suppose B2:C3 contains \left[ \! \begin{array}{cc} 3 & 5 \\ 4 & 7 \end{array} \! \right] and E2:E3 contains \left[ \! \begin{array}{c} 3 \\ 2 \end{array} \! \right]. If you highlight F4:G5, enter =B2:C3–E2:E3 and then press Crtl-Shft-Enter, F4:G5 will contain \left[ \! \begin{array}{cc} 0 & 2 \\ 2 & 5 \end{array} \! \right].

Real Statistics Data Analysis Tool: The Real Statistics Resource Pack provides the Matrix Operations data analysis tool, which supports a number of matrix operations. These work just like the corresponding worksheet array functions TRANSPOSE, MINVERSE, etc., except that you don’t need to specify the shape and size of the matrix since these are determined automatically from the shape and size of the input matrix.

Example 2: Perform all the operations supported by the Matrix data analysis tool for the data in Figure 1.

Matrix operations Excel data

Figure 1 – Data for Example 2

After pressing Ctrl-m and selecting the Matrix option, a dialog box appears. Click on the options shown in Figure 2 when the dialog box appears.

Matrix operations dialog box

Figure 2 – Dialog box for Example 2

The result is shown in Figure 3 (slightly reformatted).

Matrix operations Excel tool

Figure 3 – Output from Matrix data analysis tool

Definition 6: Vectors X1, …, Xk of the same size and shape are independent if for any scalar values b1, … bk, if b1 X1 +⋯+ bk Xk = 0, then b1 = … = bk  = 0.

Vectors X1, …, Xk are dependent if they are not independent, i.e. there are scalars b1, … bk, at least one of which is non-zero, such that b1 X1 +⋯+ bk Xk = 0. 

Observation: If X1, …, Xk  are independent, then Xj ≠ 0 for all j.

Property 6: X1, …, Xk  are dependent if and only if at least one of the vectors can be expressed as a linear combination of the others.

Proof: Suppose X1, …, Xk  are dependent. Then there are scalars b1, … bk, at least one of which is non-zero such that b1 X1 +⋯+ bk Xk = 0. Say bi ≠ 0. Then

image9047

Now suppose that Xi = \sum_{j \neq i} b_j X_j. Then b1 X1 +⋯+ bk Xk = 0, where bi = -1, and so X1, …, Xk  are dependent.

Definition 7: The dot product of two vectors X = [xi] and Y = [yj] of the same shape is defined to be the scalar

Dot product

Observation: If X and Y are n × 1 column vectors, then X∙Y XT= YTX. Also ||X|| = √X·X.

Excel Function: If R1 is the range containing the data in X and R2 is the range containing the data in Y then X · Y = SUMPRODUCT(R1, R2).

Definition 8: Two non-null vectors of the same shape are orthogonal if their dot product is 0.

12 Responses to Matrix Operations

  1. Jonathan Bechtel says:

    Hi Charles,

    I’m not sure how you arrive at the a + c, b + d, -a + 2c, and -b + 2d figures for the inverse matrix. I’m looking at the algebraic expression you wrote to arrive at it but the light bulb isn’t turning on, sorry.

    I know I can plug those in when determining one for a 2×2 matrix but it’d be nice to know in case I need to find the inverse of a larger one.

    • Charles says:

      Jonathan,
      I just used matrix multiplication and the fact that the product of the two 2×2 matrices must be the 2×2 identity matrix, which means that the values on the main diagonal are 1’s and the values off this diagonal are 0’s.
      Charles

  2. Jonathan Bechtel says:

    Hi Charles,

    Question:

    What is I2? You mention it in example 1 but I’m not clear what it means or how you arrived at the [1, 0, 0, 1] figures.

    Thank you.

  3. Adriana says:

    Dear Charles:

    Thanks for your program! it has been very useful. I have a question. When I use the matrix operations eigenvalues/eigenvectors on the same matrix of your example:
    3 -4 9
    5 2 3
    7 1 6

    I get the following result:

    Eigenvalues/vectors (QR Factorization)

    11,72209274 -1,651750232 0,929657496
    0,511963902 -0,38469524 -0,768051128
    0,482584475 -0,610875362 0,627649199
    0,71063717 0,691983283 0,127098183
    1,79176E-12 6,03212E-12 3,84593E-12
    1,06581E-14 6,881844823 4,104277377

    the first row 1×2 is the eigenvalues, next is the 3×3 matrix of eigenvectors, but next there are 2 new 1×3 rows, what are they?

    Thanks a lot!

  4. Fran says:

    the matrix operation doesn’t work on mac!

  5. moham.karta says:

    Good Article

  6. George F says:

    When I try to create the output from Matrix Operations I get the error “Compile error in hidden module: frmMatrix”. The Addin is saved in my Downloads folder but it opens, it just doesn’t compute. I do not have Admin authority and frequently IM blocks access to certain functions / executions. Am I missing something or could this function be blocked?

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