Matrix Identities (also Ring Identities)

This page assumes you are familiar with the terminology given in the Matrix Review page.

There are lots of useful matrix identities. Some are well known, some are not. Some are basic, some are derived.

(Skip down to the general Matrix identities)

The general form here is a statement of an identity, followed by the proof of that identity.

Math Geek Note: The identities here are general, in that they don't rely on the matrix being a matrix of real numbers. All I assume that the matrix elements are a ring with identity. I believe that the resulting matrices also form a ring. Therefore they are Block Matrix Identities, they are partitioned matrix identities, and they are identities for rings with identity, and they do also work for matrices of scalars. For those that have forgotten the ring axioms, or never played with ring theory, here are the basics: (: Unless I've gotten this wrong... :) Elements where addition is associative and communititive, Multiplication is associative, Additive and multiplicitive identities exist, additive and multiplicitive inverses exist, and multiplication distributes (on either side) over addition,

Notation: Note that A' here means A transpose, and that /A = inverse of A. I've waffled a great deal on the issue of how to represent matrix inverse... and the current deciding factor was that MATLAB does it this way. (And it is hopefully not too confusing that A/B is A*/B (Right multiplication by the multiplicitive inverse)

All the identities assume that the matrices in them are of appropriate sizes. They have to be the same size to add, they have to be conformable to multiply. (I.E. if C=AB, and C is n by m, A is n by k, and B is k by m for some k) They have to be square and have inverses if inverses are used. The inverses here are true inverses, while SOME of the same identities hold for psudo-inverses, there will be no attempt to deal with psudo inverses on this page.

Formatting note: Proofs on these pages are often have the form:

An expression = another expression   # Random commentary
              = something else

        Proof line of first item  [ With justifications ]
        and more lines if needed  [ and more justifications ]
       Q.E.D.

        Proof of the second item  [ Second item's justifications. ]
       Q.E.D.

Basics

Basic Definitional

Addition

A+B = B+A            Addition commutes.
A+0 = A = 0+A        There is an additive Identity.
                            (0 is used to mean identity for addition)
A-A = 0              Additive inverses exist
A-B = A + -B         Subtraction is just shorthand for adding an inverse.
-(-A)  = A           Additive inverse of an additive inverse is the original
-0 = 0               Additive identity is its own addive inverse.
-(A+B) = (-A)+(-B)   negation distributes over addition.

(A+B)+C = A+(B+C)    Addition is associative
                            Therefore we normally just write A+B+C
                            without worrying about the grouping.

We write A B or AB to mean A multiplied by B

Multiplication

A / B = A (/B)       "Division" is just shorthand for multiplying by a multiplicitive inverse
A I = A = I A        There is a multiplicative Identity.
                            (I is used to mean identity for multiplication)
A / A = I            Multiplicitive inverses exist.
//A = A              Multiplicitive inverse of a Multiplicitve inverse is the original
/I = I               Multiplicitive identity is its own multiplicitive inverse.

A(BC) = (AB)C        Therefore we normally just write ABC without worrying
                            about the grouping.

A(-B) = -(AB) = (-A)B
    Note:  AB Need not be equal to BA

Joint

A(C+D) = AC + AD    multiplication distributes over addition.
(C+D)A = CA + DA    (For either pre or post multiplication.)

 -/A  = /-A

/(AB)  = /B/A      Inverse distributes over multiplication, IN THE REVERSE ORDER.
                       [ Assuming inverses of A and B exist]

We will write  //A   for /(/A)

Matrix Definitions


I will write A' to mean transpose(A)
I will write A'' to mean (A')'

A'' = A
0'  = 0 [ Assuming that 0 is square ]
I'  = I

A is symmetric means A = A'

The transpose of a real number is that number. (scalars are symmetric)

(A+B)' = A' + B'    transpose distributes over addition.

 (AB)' =  B'A'      transpose distributes over multiplication 
                           WITH A CHANGE IN ORDER.

Derived

subtraction

A-B = -(B-A)

    A-B  = --( A -   B)  [ --x = x ]
         = --( A +  -B)  [ Definition of subtraction. ]
         = - (-A + --B)  [ Additive inverse distributes over addition ]
         = - (-A +   B)  [ --x = x ]
         = - ( B +  -A)  [ Addition commutes. ]
         = - ( B -   A)  [ Definition of subtraction ]
        Q.E.D.

multiplicitive inverse

(Using A' to mean A transpose /A to mean mulitplicitive
inverse of A)
/(ABC) = /C/B/A

       /(ABC) = /((AB)C)    [ Regroup    ]
              = /C/(AB)     [ /(xy)=/y/x ]
              = /C(/B/A)    [ /(xy)=/y/x ]
              = /C/B/A      [ Regroup    ]

Matrix Derived

(/A)' = /(A')
     By definition, the inverse of X' is /(X') and we can reason
           X/X   =  /XX    = I   [Definition of inverse]
          (X/X)' = (/XX)'  = I'  [Transpose of equals is equal]
          (/X)'X'= X'(/X)' = I   [I' = I, (AB)' = B'A']
     Therefore the inverse of X' is also (/X)' Since it is a
     right inverse and a left inverse.
     But, /(X') is the inverse.   Therefore,   /(X') = (/X)'
Therefore I will write /X' without worrying about grouping.


(ABC)' = C'B'A'

       (ABC)' = ((AB)C)'    [ Regroup  ]
              = C'(AB)'     [(AB)'=B'A']
              = C'(B'A')    [(AB)'=B'A']
              = C'B'A'      [ Regroup  ]

Symmetric Matrices

A'BA is symmetric if B is.

          B  =   B'     [definition of "B is symmetric"]
        A'B  = A'B'     [ Premultiply both sides by A']
        A'BA = A'B'A    [Postmultiply both sides by A ]
        A'BA = A'B'A''  [A=A'']
        A'BA = (A'BA)'  [(ABC)'=C'B'A')]

therefore A'BA is symmetric (Definition of symmetric)


ABA' is symmetric if B is.

        B = B'          [B is symmetric]
        ABA' = AB'A'    [B=B' from line above]
        ABA' = A''B'A'  [A=A'']
        ABA' = (ABA')'  [(ABC)'=C'B'A')]

therefore ABA' is symmetric (Definition of symmetric)

Since the Identity is symmetric, substituting I for B in the above, and using the identity that IA=A gives

A'A is symmetric
AA' is symmetric.

A+A' is symmetric. # (Assuming A is square)

     (A+A') =
            = (A'+A)     [ addition commutes ]
            = (A'+A'')   [ x = x'' ]
            = (A+A')'    [ (x'+y') = (x+y)' ]

Therefore it is equal to its own transpose, it is therefore symmetric.


A symmetric matrix +- a symmetric matrix is symmetric.

        A=A', B=B'     [ Definition of symmetric ]
        A+B  = A'+B'   [ from each being symmetric ]
        A+B  = (A+B)'  [ (A+B)' = A'+B' ]

It is equal to its own transpose, it is therefore symmetric.

An inverse of a symmetric matrix is symmetric.

     A   /A   = I                [ Definition of inverse ]
    (A   /A)' = I'               [ Transpose of both sides ]
    (A   /A)' = I                [ I' = I ]
   (/A)'  A'  = I                [ (xy)' = y'x' ]
   (/A)'  A'' = I                [ A is symmetric, so A = A' ]
   (/A)'  A   = I                [ x'' = x ]
   (/A)'  A/A = /A               [ Post multiply both sides by /A ]
   (/A)'   I  = /A               [ x/x = I ]
   (/A)'      = /A               [ xI = x ]

So the inverse of A is also equal to its own transpose, so it must also be symmetric.


General Matrix Identities [and Ring Identities]

A + B =  A ( /A  +    /B    ) B
      =  B ( /A  +    /B    ) A
      =  A ( /A  +  /A B /A ) A  # (Does not require B to be invertable)
      =  B ( /B  +  /B A /B ) B  # (Does not require A to be invertable)

      A + B =        B  + A             [ Addition commutes ]
            =   I    B  + A    I        [ Ix = x = xI ]
            = A   /A B  + A /B   B      [ x/x = I = /xx ]
            = A ( /A B  +   /B   B)     [ Left distribution ]
            = A ( /A    +   /B ) B      [ right distribution ]
           Q.E.D.

      A + B =  B        +       A       [ Addition commutes ]
            =  B    I   +  I    A       [ Ix = xI = x ]
            =  B   /AA +  B/B   A       [ x/x = I = /xx ]
            =  B ( /AA +   /B   A)      [ Left distribution ]
            =  B ( /A  +   /B ) A       [ Right distribution ]
           Q.E.D.

      A + B =   A  I  +  I  B  I        [ Ix = xI = x ]
            =   A /AA + A/A B /AA       [ I = /xx = x/x ]
            = A ( /AA +  /A B /AA )     [ left distribution ]
            = A ( /A  +  /A B /A  ) A   [ right distribution ]
           Q.E.D.

      A + B =    B     +     A          [ addition commutes ]
            =    B  I  +  I  A  I       [ Ix = xI = x ]
            =    B /BB + B/B A /BB      [ I = /xx = x/x ]
            =  B ( /BB +  /B A /BB )    [ left distribution ]
            =  B ( /B  +  /B A /B  ) B  [ right distribution ]
           Q.E.D.


I + A =   (  I  +     /A  ) A
      = A (  I  +     /A  )
      = A ( /A  +  /A /A  ) A
      = A ( /A  +  /(AA)  ) A

      I + A =     A  +  I                [ Addition commutes ]
            =   I A  +  I                [ Ix = x ]
            =   I A  + /AA               [ I = /xx ]
            = ( I    + /A) A             [ right distribution]
           Q.E.D.

      I + A =     A   +  I               [ addition commutes ]
            =     A I +  I               [ xI = x ]
            =     A I + A /A             [ I = x/x ]
            = A (   I +   /A )           [ Left distribution ]
           Q.E.D.

      I + A =  A        +  I             [ Addition commutes ]
            =  A    I   +  I     I       [ Ix = xI = x ]
            =  A   /AA  + A/A /A   A     [ I = /xx = x/x ] 
            =  A ( /AA  + A/A /A   A )   [ Left distribution ]
            =  A ( /A   +  /A /A ) A     [ Right distribution ]
           Q.E.D.

      I + A =  A        +  I             [ Addition commutes ]
            =  A    I   +  I      I      [ Ix = xI = x ]
            =  A   /AA  + A/A  /A    A   [ I = /xx = x/x ] 
            =  A ( /AA  + A/A  /A    A ) [ Left distribution ]
            =  A ( /A   +  /A  /A )  A   [ Right distribution ]
            =  A ( /A   + /(A   A) ) A   [ /(xy) = /y/x ]
           Q.E.D.


/A + /B = /A ( A  +     B   ) /B
        = /B ( A  +     B   ) /A
        = /A ( A  +  A /B A ) /A
        =    ( I  +    /B A ) /A
        = /A ( I  +  A /B   )
        = /B ( B  +  B /A B ) /B
        =    ( I  +    /A B ) /B
        = /B ( I  +  B /A   )

        /A + /B =        /B + /A               [ Addition commutes ]
                =     I  /B + /A  I            [ Ix = x = xI ]
                =    /AA /B + /A B/B           [ x/x = I = /xx ]
                = /A ( A /B +    B/B)          [ Left Distribution ]
                = /A ( A    +    B  ) /B       [ Right distribution ]
               Q.E.D.

        /A + /B =   /B     +     /A            [ Addition commutes ]
                =   /B  I  +  I  /A            [ Ix = x = xI ]
                =   /B A/A + /BB /A            [ x/x = I = /xx  ]
                = /B ( A/A +   B/A )           [ Left distribution ]
                = /B ( A   +   B   ) /A        [ Right distribution ]
               Q.E.D.

        /A + /B =     /A  I  +  I  /B  I       [ Ix = xI = x ]
                =     /A A/A + /AA /B A/A      [ /xx = x/x = I ]
                = /A (  A/A +    A /B A/A)     [ Left distribution ]
                = /A (  A   +    A /B A  ) /A  [ Right distribution ]
               Q.E.D.

        /A + /B = /A (  A   +    A /B A  ) /A    [ previous identity  ]
                = (  /A A   + /A A /B A  ) /A    [ Left distribution  ]
                = (    I    +   I  /B A  ) /A    [ /x x = I           ]
                = (    I    +      /B A  ) /A    [ I x = x            ]
               Q.E.D.

        /A + /B = /A (  A    +    A /B A    ) /A [ previous identity  ]
                = /A (  A /A +    A /B A /A )    [ Right distribution ]
                = /A (   I   +    A /B  I   )    [ x /x = I           ]
                = /A (   I   +    A /B      )    [ x I = x            ]

        /A + /B = /B       +     /A            [ Addition commutes ]
                = /B    I  +  I  /A  I         [ Ix = x = xI ]
                = /B   B/B + /BB /A B/B        [ x/x = I = /x ]
                = /B ( B/B +   B /A B/B )      [ Left distribution ]
                = /B ( B   +   B /A B   ) /B   [ Right distribution ]       
               Q.E.D.

        /A + /B = /B (  B   +    B /A B  ) /B    [ previous identity ]
                = (  /B B   + /B B /A B  ) /B    [ Left distribution ]
                = (    I    +   I  /A B  ) /B    [ /x x = I          ]
                = (    I    +      /A B  ) /B    [ I x = x           ]
               Q.E.D.

        /A + /B = /B (  B   +    B /A B  ) /B    [ previous identity  ]
                = /B (  B /B +    B /A B /B )    [ Right distribution ]
                = /B (   I   +    B /A  I   )    [ x /x = I           ]
                = /B (   I   +    B /A      )    [ x I = x            ]
               Q.E.D.


/(A+B) =      /A /(/A + /B) /B
       =      /B /(/A + /B) /A
       = /A - /A /(/A + /B) /A
       = /B - /B /(/A + /B) /B

      /(A + B) =  /( B         +        A   )     [ Addition commutes ]
               =  /( B     I   +  I     A   )     [ Ix = xI = x ]
               =  /( B    /AA +  B/B    A   )     [ x/x = I = /xx ]
               =  /( B  ( /AA +   /B    A)  )     [ Left distribution ]
               =  /( B  ( /A  +   /B )  A   )     [ Right distribution ]
               =    /A /( /A +    /B ) /B         [ /(abc) = /c/b/a ]
           Q.E.D.

      /(A + B) = /(         B + A         )       [ Addition commutes ]
               = /(      I  B + A  I      )       [ Ix = x = xI ]
               = /(     A/A B + A /BB     )       [ x/x = I = /xx ]
               = /( A  ( /A B +   /BB)    )       [ Left distribution ]
               = /( A  ( /A   +   /B )  B )       [ right distribution ]
               =   /B /( /A   +   /B ) /A         [ /(abc) = /c/b/a ]
              Q.E.D.

       /(A+B) =         /(//A + //B)     [ //x = x ]
              = /A - /A /( /A +  /B) /A  [ /(/x+/y) = x /(x+y) y [x=/A, y=/B ]
             Q.E.D.

       /(A+B) = /(//A+//B)               [ //x = x ]
              = /B - /B /(/A+/B) /B      [ /(/x+/y) = x /(x+y) y [x=/B, y=/A ]
             Q.E.D.



I + /A =    ( I +    A ) /A
       = /A ( I +    A )
       = /A ( A +  A A ) /A

        I + /A =  /A +   I                 [ Addition commutes ]
               =  /A +  A /A               [ I = x/x ]
               = ( I +  A ) /A             [ Right distribution ]
              Q.E.D.

        I + /A =      /A +   I             [ Addition commutes ]
               =      /A + /A A            [ I = /xx ]
               = /A (  I +    A )          [ Left distribution ]
              Q.E.D.

        I + /A =     /A     +  I           [ Addition commutes ]
               =     /A  I  +  I   I       [ Ix = x = xI ]
               =     /A A/A + /AA A/A      [ /xx = I = x/x ]
               = /A (   A/A +   A A/A)     [ Left distribution ]
               = /A (   A   +   A A  ) /A  [ Right distribution ]
              Q.E.D.


/(I  +  /A B) = /(A + B) A

	/(I + /AB) = /(   I  + /AB      )      [ Identity ]
                   = /( /A A + /AB      )      [ I = /xx ]
                   = /( /A(A +   B)     )      [ Left Distribution ]
                   =     /(A +   B) //A        [ /(xy)= /y/x ]
                   =     /(A +   B)   A        [ //x = x ]
                  Q.E.D.


/(I + AB) = /(/A + B) /A

	/(I + AB)  = /(    I   +  AB     )     [ Identity ]
                   = /(  A  /A +  AB     )     [ I = /xx ]
                   = /(  A (/A +   B)    )     [ Left Distribution ]
                   =      /(/A +   B) /A       [ /(xy)= /y/x ]
                  Q.E.D.


# and the mixed:
A+/B =  A ( /A +    B   ) /B
     = /B ( /A +    B   )  A
     =  A ( /A + /A/B/A )  A
     = /B (  B +  B A B ) /B   # (Does not require A to be invertable)

     A + /B =        /B  +  A             [ Addition commutes ] 
            =     I  /B  +  A  I          [ Ix = x = xI ]
            =    A/A /B  +  A B/B         [ /xx = I = x/x ]
            = A ( /A /B  +    B/B )       [ Left distribution ]
            = A ( /A     +    B   ) /B    [ Right distribution ]
           Q.E.D.

     A + /B = /B       +     A            [ Addition commutes ]
            = /B    I  +  I  A            [ Ix = x = xI ]
            = /B   /AA + /BB A            [ /xx = I = x/x ]
            = /B ( /AA +   B A )          [ Left distribution ]  
            = /B ( /A  +   B   ) A        [ Right distribution ]
           Q.E.D.

     A + /B =    A  I  +  I  /B  I        [ Ix = xI = x ]
            =    A /AA + A/A /B /AA       [ I=/xx=x/x ]
            =  A ( /AA +  /A /B /AA )     [ Left distribution ]
            =  A ( /A  +  /A /B /A  ) A   [ Right distribution ]
           Q.E.D.

     A + /B =        /B +     A           [ Addition commutes ]
            =     I  /B +  I  A  I        [ Ix = x = xI ]
            =    /BB /B + /BB A B/B       [ /xx = I = x/x ]
            = /B ( B /B +   B A B/B )     [ Left distribution ]
            = /B ( B    +   B A B   ) /B  [ Right distribution ]
           Q.E.D.


A-B = - A ( /A - /B ) B
    = - B ( /A - /B ) A
    =   A ( /B - /A ) B
    =   B ( /B - /A ) A

    A - B =           A  + (-B)               [ Definition of - ]
          =         (-B) +   A                [ Addition commutes ]
          =      I  (-B) +   A  I             [ Ix = x = xI ] 
          =     A/A (-B) +   A /(-B)(-B)      [ x/x = I = /xx ]
          =   A (/A (-B) +     /(-B)(-B) )    [ Left distribution ]
          =   A (/A      +     /(-B) ) (-B)   [ Right distribution. ]
          = - A (/A      +     /(-B) )   B    [ x(-y) = -xy ]
          = - A (/A      +      -/B  )   B    [ /-x = -/x ]
          = - A (/A      -       /B  )   B    [ Definition of subtraction ]
         Q.E.D.

    A - B =             A  +  (-B)           [ Definition of subtraction ]
          =       I     A  +  (-B)  I        [ Ix = x = xI ]
          =   (-B)/(-B) A  +  (-B) /AA       [ x/x = I = /xx ]
          = (-B) (/(-B) A  +       /AA)      [ Left distribution ]
          = (-B) (/(-B)    +       /A ) A    [ Right distribution ]
          =  -B  (/(-B)    +       /A ) A    [ Regroup ]
          =  -B  ( -/B     +       /A ) A    [ /(-x) = -/x ]
          =  -B  (  /A     +      -/B ) A    [ Addition commutes ]
          =  -B  (  /A     -       /B ) A    [ Definition of subtraction ]
         Q.E.D.

    A - B = - A   (/A - /B)  B               [ x-y = -x(/x-/y)y ]
          =   A (-(/A - /B)) B               [ -xyz = x(-y)z ]
          =   A   (/B - /A)  B               [ -(x-y) = y-x ]
         Q.E.D.

    A - B = -B   (/A - /B)  A                [ x-y = -y(/x-/y)x ]
          =  B (-(/A - /B)) A                [ -xyz = x(-y)z ]
          =  B   (/B - /A)  A                [ -(x-y) = y-x ]
         Q.E.D.


AB + BC = A (/AB + B/C) C  # (Does not require B to be invertable, or even square)

        AB+BC =       BC + AB        [ addition commutes ]
              =    I  BC + AB  I     [ Ix = x = xI ]
              =   A/A BC + AB /CC    [ x/x = I = /xx ]
              = A (/A BC +  B /C)    [ left distribution ]
              = A (/A B  +  B /C) C  [ Right distribution ]
             Q.E.D.


A + AB = (A + A/B) B  # (Does not require A to be invertable, or even square)

       A+AB =   AB +  A        [ addition commutes  ]
            =   AB +  A   I    [ x = xI             ]
            =   AB +  A /B  B  [ x/x = I = /xx      ]
            = ( A  +  A /B) B  [ Right distribution ]
             Q.E.D.


/(/A+/B) =     A /(A+B) B
         =     B /(A+B) A
         = A - A /(A+B) A  # These last two identities were extensively used by
         = B - B /(A+B) B  # Fuzhen Zhang in his book "Matrix Theory"

       /(/A + /B) =  /( /B         +        /A   )     [ Addition commutes ]
                  =  /( /B     I   +  I     /A   )     [ Ix = xI = x ]
                  =  /( /B    A/A  + /BB    /A   )     [ x/x = I = /xx ]
                  =  /( /B  ( A/A +    B    /A ) )     [ Left distribution ]
                  =  /( /B  ( A    +   B )  /A   )     [ Right distribution ]
                  =    //A /( A  +     B ) //B         [ /(abc) = /c/b/a ]
                  =      A /( A  +     B )   B         [ //x = x ]
                 Q.E.D.

       /(/A + /B) =  /( /A         +        /B   )     [ Addition commutes ]
                  =  /( /A     I   +  I     /B   )     [ Ix = xI = x ]
                  =  /( /A    A/A  + /BB    /B   )     [ x/x = I = /xx ]
                  =  /( /A  ( A/A  +   B    /B ) )     [ Left distribution ]
                  =  /( /A  ( A    +   B )  /B   )     [ Right distribution ]
                  =    //B /( A    +   B ) //A         [ /(abc) = /c/b/a ]
                  =      B /( A    +   B )   A         [ //x = x ]
                 Q.E.D.

                           A = A             [ identity ]
                     I     A = A             [ Ix = x ]
                (A+B)/(A+B)A = A             [ x/x = I ]
         A/(A+B)A + B/(A+B)A = A             [ left Distribution ]
                    B/(A+B)A = A - A/(A+B)A  [ Subtract A/(A+B)A from both sides ]
                    /(/A+/B) = A - A/(A+B)A  [ y/(x+y)x = /(/x+/y) ]
                            Q.E.D.

                           B = B             [ identity ]
                     I     B = B             [ Ix = x ]
                (A+B)/(A+B)B = B             [ x/x = I ]
         A/(A+B)B + B/(A+B)B = B             [ Left distribution ]
         A/(A+B)B            = B - B/(A+B)B  [ Subtract B/(A+B)B from both sides ]
         /(/A+/B)            = B - B/(A+B)B  [ x/(x+y)y = /(/x+/y) ]
                            Q.E.D.


/(I+/A) =       /(I+A) A
        =     A /(I+A)
        = I -   /(I+A)
        = A - A /(I+A) A

        /( I + /A ) = /( /I + /A )            [ I = /I ]
                    = I /(I + A) A            [ /(/x+/y) = x /(x + y) y ]
                    =   /(I + A) A            [ Ix = x ]
                   Q.E.D.

        /( I + /A ) = /( /I + /A )            [ I = /I ]
                    = A /(I + A) I            [ /(/x+/y) = y /(x + y) x ]
                    = A /(I + A)              [ Ix = x ]
                   Q.E.D.

        /( I + /A ) =       /( /I + /A )      [ I = /I ]
                    = I - I /(  I +  A ) I    [ /(/x+/y) = x - x/(x+y)x ]
                    = I -   /(  I +  A )      [ Ix = I; xI = I ]
                   Q.E.D.

        /( I + /A ) =       /( /I + /A )      [ I = /I ]
                    = A - A /(  I +  A ) A    [ /(/x+/y) = y - y/(x+y)y ]
                   Q.E.D.


/(I+A) =         /(I + /A) /A
       =      /A /(I + /A)
       =  I -    /(I + /A)
       = /A - /A /(I + /A) /A
       =      /A /(I +  A)  A
       =       A /(I +  A) /A

       /(I+A) = /(A  (/I + /A)  I)       [ x+y = y(/x+/y)x ]
              =  /I /(/I + /A) /A        [ /(xyz) = /z/y/x ]
              =     /( I + /A) /A        [ /I = I, Ix=x ]
             Q.E.D.

       /(I+A) = /(I  (/I + /A)  A)       [ x+y = x(/x+/y)y ]
              =  /A /(/I + /A) /I        [ /(xyz) = /z/y/x ]
              =  /A /( I + /A)           [ /I=I, xI=I ]
             Q.E.D.

       /(I+A) =         /(//I + //A)     [ //x = x ]
              = /I - /I /( /I +  /A) /I  [ /(/x+/y) = x /(x+y) y [x=/I, y=/A ]
              =  I -    /(  I +  /A)     [ /I=I, Ix=xI=I ]
             Q.E.D.

       /(I+A) =         /(//I + //A)     [ //x = x ]
              = /A - /A /( /I +  /A) /A  [ /(/x+/y) = x /(x+y) y [x=/A, y=/I ]
              = /A - /A /(  I +  /A) /A  [ /I=I ]
             Q.E.D.

       /(I+A) = /A /( I + /A)      [ /(I+x) = /x /(I + /x) ]
              = /A /( I +  A) A    [ /(I+/x) = /(I + x) x ]
             Q.E.D.

       /(I+A) =    /( I + /A) /A    [ /(I+x) = /(I + /x) /x ]
              =  A /( I +  A) /A    [ /(I+/x) = x /(I + /x) ]
             Q.E.D.


/(I-A) = /A /( I - A )  A
       =  A /( I - A ) /A

       /(I-A) =      /( I + -A )        [ x-y = x+-y ]
              = (-A) /( I + -A) /(-A)   [ /(I+x) = x/(I+x)/x ]
              =   A  /( I -  A) /  A    [ x+-y=x-y, /-x = -/x, (-x)y(-z) = xyz ]
             Q.E.D.

       /(I-A) =      /( I + -A )        [ x-y = x+-y ]
              = /(-A) /( I + -A) (-A)   [ /(I+x) = x/(I+x)/x ]
              = /  A  /( I -  A)   A    [ x+-y=x-y, /-x = -/x, (-x)y(-z) = xyz ]
             Q.E.D.


B /( I + AB )   = /A /( I + /(AB) )
                = /A /( I +   AB  ) AB

   B /( I + AB)  =  B  /(AB) /(I+/(AB))     [ /(I+x) = /x /(I + /x) ]
                 =  B / B /A /(I+/(AB))     [ /(xy) = /y /x ]
                 =        /A /(I+/(AB))     [ x /x = I ]
                Q.E.D.
                 =        /A /(I+  AB) AB   [ /(I+/x) = /(I+x) x ]
                Q.E.D.


/( I + AB ) A =      /( I + /(AB) ) /B
              =  A B /( I +   AB  ) /B

   /( I + AB ) A =    /(I + /(AB)) /(AB)  A  [ /(I+x) = /(I+/x) /x ]
                 =    /(I + /(AB)) /B /A  A  [ /(xy) = /y/x ]
                 =    /(I + /(AB)) /B        [ /x x = I, x I = x ]
                 = AB /(I +   AB ) /B        [ /(I+/x) = x /(I+x) ]
                Q.E.D.

    /( I + AB) A = /(I+/(AB)) /(AB) A       [ /(I+x) = /(I+/x) /x ]
                 = /(I+/(AB)) /B /A A       [  /(xy) = /y /x      ]
                 = /(I+/(AB)) /B            [  /x  x = I          ]
               Q.E.D.


/A - /B = /A (B-A) /B # (Sherman-Morrison identity in disguise)
        = /B (B-A) /A

         /A - /B = /A      -      /B  [     Identity       ] 
                 = /A   I  -  I   /B  [    Ix = x = xI     ]
                 = /A  B/B - /AA  /B  [   /xx = I = x/x    ]
                 = /A (B/B -   A  /B) [ left distribution  ]
                 = /A (B   -   A) /B  [ right distribution ]
               Q.E.D.

         /A - /B = /A      -      /B  [     Identity       ] 
                 =  I  /A  -  /B  I   [    Ix = x = xI     ]
                 = /BB /A  -  /B A/A  [   /xx = I = x/x    ]
                 = /B (B/A -   A  /A) [ left distribution  ]
                 = /B (B   -   A) /A  [ right distribution ]
               Q.E.D.



/(/A-/B) = B /(B-A) A 
         = A /(B-A) B

         /(/A-/B) = /(       /A - /B      ) [     Identity      ]
                  = /( /A  (  B -  A ) /B ) [ Previous identity ]
                  =   //B /(  B -  A )//A   [ /(abc) = /c/b/a   ]
                  =     B /(  B -  A )  A   [    //x = x        ]
               Q.E.D.

         /(/A-/B) = /(       /A - /B      ) [     Identity      ]
                  = /( /B  (  B -  A ) /B ) [ Previous identity ]
                  =   //A /(  B -  A )//B   [ /(abc) = /c/b/a   ]
                  =     A /(  B -  A )  B   [    //x = x        ]
               Q.E.D.

A + A /B A = A /B (  A +  B )
           =      (  A +  B ) /B A
           = A    ( /A + /B )    A

     A + A /B A = A      ( /A + /B )     A  [ Previous identity ]
                = A ( /B (  A +  B ) /A) A  [ /x + /y = /y (x + y) /x ]
                = A   /B (  A +  B )        [ Regroup and x/x = I, Ix = x ]
               Q.E.D.

     A + A /B A = A     ( /A + /B )    A  [ Previous identity ]
                = A ( /A ( A + B ) /B) A  [ /x + /y = /x (x + y) /y ]
                =        ( A + B ) /B  A  [ Regroup and x/x = I, Ix = x ]
               Q.E.D.

      A + A /B A = A (  I  + /BA)         [ left distribution ]
                 = A ( /AA + /BA)         [ I = x'x ]
                 = A ( /A  + /B ) A       [ Right distribution ]
               Q.E.D.


A - A /B A = A /B (  B -  A )
           =      (  B -  A ) /B A
           = A    ( /A - /B )    A

         A - A /B A = A /B  B - A /B A    [ x = x /y y ]
                    = A /B (B - A)        [ xy-xz = x(y-z) ]
                   Q.E.D.

         A - A /B A =       A - A /B A    [ Identity ]
                    =  B /B A - A /B A    [ y = x /x y ]
                    = (B - A) /B A        [ yx-zx = (y-z)x ]
                   Q.E.D.

         A - A /B A = A (  I - /B A )     [ Left distribution ]
                    = A ( /A - /B   ) A   [ Right distribution ]
                  Q.E.D.


A (/A + /B)   =    (A+B) /B
  (/A + /B) B = /A (A+B)

       A (/A+/B)   = A(/A (A+B) /B)   [ x+y = x ( /x + /y ) y ]
                   = A /A (A+B) /B    [ Regroup               ]
                   =      (A+B) /B    [ Cancel A/A            ]
                Q.E.D.

         (/A+/B) B =  (/A (A+B) /B) B [ x+y = x ( /x + /y ) y ]
                   =   /A (A+B) /B  B [ Regroup               ]
                   =   /A (A+B)       [ Cancel /BB            ]
                Q.E.D.


A (/A+ B)   =   ( A + /B) B
  ( A+/B) B = A (/A +  B)

       A (/A+B) = A (B+ /A)      [ Addition commutes ]
                =   AB+A/A       [ Left distribution ]
                =   AB+ I        [ x/x = I ]
                =   AB+/BB       [ I = /xx ]
                =  (A +/B) B     [ Right distribution. ]
               Q.E.D.

       (A+/B) B =    AB + /BB    [ Right distribution ]
                =    AB + I      [ /xx = I ]
                =    AB + A/A    [ I = x/x ]
                = A ( B + /A)    [ Left distribution ]
                = A (/A +  B)    [ x+y = y+x ]
               Q.E.D.


A /( A + B )   =     /( /A + /B ) /B
  /( A + B ) B =  /A /( /A + /B )

       A /(A+B) = A /(B (/A+/B) A)  [ x+y = y(/x+/y)x ]
                = A /A /(/A+/B) /B  [ /(xyz) = /z/y/x ]
                =      /(/A+/B) /B  [ A/A = I ]
              Q.E.D.

       /(A+B) B = /(B (/A+/B) A) B  [ x+y = y(/x+/y)x ]
                = /A /(/A+/B) /B B  [ /(xyz) = /x/y/z ]
                = /A /(/A+/B)       [ /x x = I ]
              Q.E.D.


/A (A + B)    =   ( /A + /B ) B
   (A + B) /B = A ( /A + /B )

         /A (A+B) = /AA + /AB    [ Left distribution. ]
                  =  I  + /AB    [ /xx = I ]
                  = /BB + /AB    [ I = /xx ]
                  =  (/B +/A)B   [ right distribution ]
                  =  (/A +/B) B  [ x + y = y + x ]
                Q.E.D.

         (A+B) /B = A/B + B/B    [ right distribution ]
                  = A/B +  I     [ x/x = I ]
                  = A/B + A/A    [ I = x/x ]
                  = A (/B+/A)    [ left distribution ]
                  = A (/A+/B)    [ x + y = y + x ]
                Q.E.D.


/A /(/A+/B)    =   /(A+B) B
   /(/A+/B) /B = A /(A+B)

      /A /(/A+/B) = /A A /(A+B) B   [ /(/x+/y) = x /(x+y) y ]
                  =   I  /(A+B) B   [ /x x = I ]
                  =      /(A+B) B   [ I x = x  ]
                 Q.E.D.

      /(/A+/B) /B = A /(A+B) B /B   [ /(/x+/y) = x /(x+y) y ]
                  = A /(A+B) I      [ /x x = I ]
                  = A /(A+B)        [ x I = x ]
                 Q.E.D.


A/B+I     = A(/A+/B)

       A  +  B   = A(/A+/B)B        [ x+y = x(/x+/y)y ]
      (A/B+I)B   = A(/A+/B)B        [ Distribution extracting B ]
       A/B+I     = A(/A+/B)         [ postmultiply both sides by /B]
                Q.E.D.


A + B + ACB = A(/A+/B+C)B

       A+B+ACB = A(/A+/B)B+ACB  [ x+y = x(/x+/y)y ]
               = A(/A+/B+C)B    [ Distribute from both sides ]
              Q.E.D.

--------------------- Symmetric matrix products ------------------------

IF A is symmetric, and B is symmetric, AND the their product is symmetric,
THEN A=A', B=B', AB=C, and C=C', AND
   C = C'      [ Given ]
  AB = (AB)'   [ Substituting AB for C ]
  AB = B'A'    [ basic identity of ' ]
  AB = B A     [ B=B', A=A', given ]

So... in (at least) *THAT* specific case, A and B commute.

---------------------- Orthogonal matrix notes. ---------------------------

An orthogonal matrix is one where /A = A'

/A' = A  if A is orthogonal.

     /A' = /A'  [ x = x ]
     /A' = A''  [ Def. of A orthogonal /x=x' ]
     /A' = A    [ x'' = x ]

Generating matrix identities by matrices

Since matrices are rings if their elements form a ring, we can take advantage of this to generate some non-trivial identities.

In a ring, if there is a left inverse, and a right inverse, they are formally identical... even if different derivations give you something that looks different. So for example, the right inverse of the matrix

[ A B ]
[ C D ]

is:

[     /(A - B/DC)   -/AB/(D - C/AB) ]
[ -/DC/(A - B/DC)       /(D - C/AB) ]

and the left inverse is:

[  /(A - B/DC)     -/(A - B/DC)B/D  ]
[ -/(D - C/AB)C/A   /(D - C/AB)     ]

Since the left and right inverses are formally identical, they must be element by element formally identical. So we can immediately read off the following identities from the off diagonal elements. (The diagonal elements being already identical in form they are trivially identical)

 -/A B /(D - C /A B)   =   -/(A - B /D C) B /D
                      and
 -/D C /(A - B /D C)   =   -/(D - C /A B) C /A

or  (Applying  -x = -y   =>  x = y )

 /A B /(D - C /A B)   =   /(A - B /D C) B /D
                     and
 /D C /(A - B /D C)   =   /(D - C /A B) C /A

or by applying /xyz = /z/y/x and //x = x

 (D - C /A B) /B A    =   D /B (A - B /D C)
                     and
 (A - B /D C) /C D    =   A /C (D - C /A B)

All of which might be a little hard to find directly.  :)

Other identities above could be used from there to get things like:

    (   D      - C /A B)   /B A  =   D /B   (   A      - B /D C)   [ Identity above ]
  C (/C D /B   -   /A  ) B /B A  =   D /B B (\B A \C   -   /D  ) C [ x-yzw = y(\yz\w - z)w ]
 (C  /C D /B   - C /A  )      A  =       (D  \B A \C   - D /D  ) C [ left distribution ]
  C  /C D /B A - C /A A          =        D  \B A \C C - D /D C    [ left distribution ]
        D /B A - C               =        D  /B A      -      C    [ x \x = I = \x x ]

Go to ...


This page is http://www.cc.utah.edu/~nahaj/math/matrix-identities.html
© Copyright 2003-2010 by John Halleck, All Rights Reserved.
This snapshot was last modified on August 19th, 2011
And the underlying file was last modified on January 20th, 2010