RGB/XYZ Matrices

Given the chromaticity coordinates of an RGB system ($x_r$, $y_r$), ($x_g$, $y_g$) and ($x_b$, $y_b$) and its reference white ($X_W$, $Y_W$, $Z_W$), here is the method to compute the 3 × 3 matrix for converting RGB to XYZ:

$$\left[\matrix{X\\Y\\Z}\right] = [M]\left[\matrix{R\\G\\B}\right]$$

where

$$[M] = \left[\matrix{{S_r X_r} & {S_g X_g} & {S_b X_b} \\ {S_r Y_r} & {S_g Y_g} & {S_b Y_b} \\ {S_r Z_r} & {S_g Z_g} & {S_b Z_b}}\right]$$ $$X_r = x_r / y_r$$ $$Y_r = 1$$ $$Z_r = (1 - x_r - y_r) / y_r$$ $$X_g = x_g / y_g$$ $$Y_g = 1$$ $$Z_g = (1 - x_g - y_g) / y_g$$ $$X_b = x_b / y_b$$ $$Y_b = 1$$ $$Z_b = (1 - x_b - y_b) / y_b$$ $$\left[\matrix{S_r\\S_g\\S_b}\right] = {\left[\matrix{X_r & X_g & X_b \\ Y_r & Y_g & Y_b \\ Z_r & Z_g & Z_b}\right]}^{-1} \left[\matrix{X_W\\Y_W\\Z_W}\right]$$

Implementation Notes:

  1. In order to properly use this matrix, the RGB values must be linear and in the nominal range [0.0, 1.0]. In many cases, RGB values may first need conversion (for example, dividing by 255 and then raising them to a power).
  2. Be careful that reference whites are used consistently. For example, sRGB is defined relative to a D65 reference white and ICC profiles are defined relative to a D50 reference white. Mismatched reference whites must be accounted for elsewhere, typically by using a chromatic adaptation algorithm.
  3. The inverse matrix (i.e. the matrix converting XYZ to RGB) is computed by inverting matrix [M] above.

Some Common RGB Working Space Matrices

Using the chromaticities and reference whites of common RGB working spaces, I have done the math for you to compute the RGB-to-XYZ and XYZ-to-RGB matrices. Note that these matrices are given relative to their own reference whites. If you examine the matrices for these working spaces found inside ICC profiles (through the redColorantTag, greenColorantTag and blueColorantTag), those matrices will always be relative to D50, and therefore, the colorants have been subjected to a chromatic adaptation transformation if the working space reference white is not also D50.

RGB Working Space Reference White RGB to XYZ [M] XYZ to RGB [M]-1
Adobe RGB (1998) D65 
 0.5767309  0.1855540  0.1881852
 0.2973769  0.6273491  0.0752741
 0.0270343  0.0706872  0.9911085
 
 2.0413690 -0.5649464 -0.3446944
-0.9692660  1.8760108  0.0415560
 0.0134474 -0.1183897  1.0154096
AppleRGB D65 
 0.4497288  0.3162486  0.1844926
 0.2446525  0.6720283  0.0833192
 0.0251848  0.1411824  0.9224628
 
 2.9515373 -1.2894116 -0.4738445
-1.0851093  1.9908566  0.0372026
 0.0854934 -0.2694964  1.0912975
Best RGB D50 
 0.6326696  0.2045558  0.1269946
 0.2284569  0.7373523  0.0341908
 0.0000000  0.0095142  0.8156958
 
 1.7552599 -0.4836786 -0.2530000
-0.5441336  1.5068789  0.0215528
 0.0063467 -0.0175761  1.2256959
Beta RGB D50 
 0.6712537  0.1745834  0.1183829
 0.3032726  0.6637861  0.0329413
 0.0000000  0.0407010  0.7845090
 
 1.6832270 -0.4282363 -0.2360185
-0.7710229  1.7065571  0.0446900
 0.0400013 -0.0885376  1.2723640
Bruce RGB D65 
 0.4674162  0.2944512  0.1886026
 0.2410115  0.6835475  0.0754410
 0.0219101  0.0736128  0.9933071
 
 2.7454669 -1.1358136 -0.4350269
-0.9692660  1.8760108  0.0415560
 0.0112723 -0.1139754  1.0132541
CIE RGB E 
 0.4887180  0.3106803  0.2006017
 0.1762044  0.8129847  0.0108109
 0.0000000  0.0102048  0.9897952
 
 2.3706743 -0.9000405 -0.4706338
-0.5138850  1.4253036  0.0885814
 0.0052982 -0.0146949  1.0093968
ColorMatch RGB D50 
 0.5093439  0.3209071  0.1339691
 0.2748840  0.6581315  0.0669845
 0.0242545  0.1087821  0.6921735
 
 2.6422874 -1.2234270 -0.3930143
-1.1119763  2.0590183  0.0159614
 0.0821699 -0.2807254  1.4559877
Don RGB 4 D50 
 0.6457711  0.1933511  0.1250978
 0.2783496  0.6879702  0.0336802
 0.0037113  0.0179861  0.8035125
 
 1.7603902 -0.4881198 -0.2536126
-0.7126288  1.6527432  0.0416715
 0.0078207 -0.0347411  1.2447743
ECI RGB D50 
 0.6502043  0.1780774  0.1359384
 0.3202499  0.6020711  0.0776791
 0.0000000  0.0678390  0.7573710
 
 1.7827618 -0.4969847 -0.2690101
-0.9593623  1.9477962 -0.0275807
 0.0859317 -0.1744674  1.3228273
Ekta Space PS5 D50 
 0.5938914  0.2729801  0.0973485
 0.2606286  0.7349465  0.0044249
 0.0000000  0.0419969  0.7832131
 
 2.0043819 -0.7304844 -0.2450052
-0.7110285  1.6202126  0.0792227
 0.0381263 -0.0868780  1.2725438
NTSC RGB C 
 0.6068909  0.1735011  0.2003480
 0.2989164  0.5865990  0.1144845
 0.0000000  0.0660957  1.1162243
 
 1.9099961 -0.5324542 -0.2882091
-0.9846663  1.9991710 -0.0283082
 0.0583056 -0.1183781  0.8975535
PAL/SECAM RGB D65 
 0.4306190  0.3415419  0.1783091
 0.2220379  0.7066384  0.0713236
 0.0201853  0.1295504  0.9390944
 
 3.0628971 -1.3931791 -0.4757517
-0.9692660  1.8760108  0.0415560
 0.0678775 -0.2288548  1.0693490
ProPhoto RGB D50 
 0.7976749  0.1351917  0.0313534
 0.2880402  0.7118741  0.0000857
 0.0000000  0.0000000  0.8252100
 
 1.3459433 -0.2556075 -0.0511118
-0.5445989  1.5081673  0.0205351
 0.0000000  0.0000000  1.2118128
SMPTE-C RGB D65 
 0.3935891  0.3652497  0.1916313
 0.2124132  0.7010437  0.0865432
 0.0187423  0.1119313  0.9581563
 
 3.5053960 -1.7394894 -0.5439640
-1.0690722  1.9778245  0.0351722
 0.0563200 -0.1970226  1.0502026
sRGB D65 
 0.4124564  0.3575761  0.1804375
 0.2126729  0.7151522  0.0721750
 0.0193339  0.1191920  0.9503041
 
 3.2404542 -1.5371385 -0.4985314
-0.9692660  1.8760108  0.0415560
 0.0556434 -0.2040259  1.0572252
Wide Gamut RGB D50 
 0.7161046  0.1009296  0.1471858
 0.2581874  0.7249378  0.0168748
 0.0000000  0.0517813  0.7734287
 
 1.4628067 -0.1840623 -0.2743606
-0.5217933  1.4472381  0.0677227
 0.0349342 -0.0968930  1.2884099

For those RGB working spaces that are not natively D50, the Bradford-adapted, D50 matrices are:

RGB Working Space Reference White RGB to XYZ [M] XYZ to RGB [M]-1
Adobe RGB (1998) D50 
 0.6097559  0.2052401  0.1492240
 0.3111242  0.6256560  0.0632197
 0.0194811  0.0608902  0.7448387
 
 1.9624274 -0.6105343 -0.3413404
-0.9787684  1.9161415  0.0334540
 0.0286869 -0.1406752  1.3487655
AppleRGB D50 
 0.4755678  0.3396722  0.1489800
 0.2551812  0.6725693  0.0722496
 0.0184697  0.1133771  0.6933632
 
 2.8510695 -1.3605261 -0.4708281
-1.0927680  2.0348871  0.0227598
 0.1027403 -0.2964984  1.4510659
Bruce RGB D50 
 0.4941816  0.3204834  0.1495550
 0.2521531  0.6844869  0.0633600
 0.0157886  0.0629304  0.7464909
 
 2.6502856 -1.2014485 -0.4289936
-0.9787684  1.9161415  0.0334540
 0.0264570 -0.1361227  1.3458542
CIE RGB D50 
 0.4868870  0.3062984  0.1710347
 0.1746583  0.8247541  0.0005877
-0.0012563  0.0169832  0.8094831
 
 2.3638081 -0.8676030 -0.4988161
-0.5005940  1.3962369  0.1047562
 0.0141712 -0.0306400  1.2323842
NTSC RGB D50 
 0.6343706  0.1852204  0.1446290
 0.3109496  0.5915984  0.0974520
-0.0011817  0.0555518  0.7708399
 
 1.8464881 -0.5521299 -0.2766458
-0.9826630  2.0044755 -0.0690396
 0.0736477 -0.1453020  1.3018376
PAL/SECAM RGB D50 
 0.4552773  0.3675500  0.1413926
 0.2323025  0.7077956  0.0599019
 0.0145457  0.1049154  0.7057489
 
 2.9603944 -1.4678519 -0.4685105
-0.9787684  1.9161415  0.0334540
 0.0844874 -0.2545973  1.4216174
SMPTE-C RGB D50 
 0.4163290  0.3931464  0.1547446
 0.2216999  0.7032549  0.0750452
 0.0136576  0.0913604  0.7201920
 
 3.3921940 -1.8264027 -0.5385522
-1.0770996  2.0213975  0.0207989
 0.0723073 -0.2217902  1.3960932
sRGB D50 
 0.4360747  0.3850649  0.1430804
 0.2225045  0.7168786  0.0606169
 0.0139322  0.0971045  0.7141733
 
 3.1338561 -1.6168667 -0.4906146
-0.9787684  1.9161415  0.0334540
 0.0719453 -0.2289914  1.4052427